# A classification of the division algebras that are isotopic to a cyclic
Galois field extension

S. Pumplün<EMAIL_ADDRESS>School of Mathematical Sciences
University of Nottingham
University Park
Nottingham NG7 2RD
United Kingdom

###### Abstract.

We classify division algebras that are principal Albert isotopes of cyclic
Galois field extensions of degree $n>2$ up to isomorphisms. We achieve a
“tight” classification when the cyclic Galois field extensions is cubic. The
classification is “tight” in the sense that the list of algebras has features
that make it easy to distinguish non-isomorphic ones. Rough classification
results on regular algebras that are principal Albert isotopes of Galois
$C_{n}$-algebras are obtained as well; the principal Albert isotopes of
$C_{3}$-Galois algebras are again classified up to isomorphisms.

###### Key words and phrases:

Nonassociative division algebras; isotopy; cyclic field extension;
classification.

###### 2020 Mathematics Subject Classification:

Primary: 17A35; Secondary: 17A60, 17A99

## Introduction

Petersson’s approach to classify the two-dimensional algebras over a field up
to isomorphism works for any base field and distinguishes between regular and
singular algebras [19]. It achieves a full classification of all
nonassociative two-dimensional algebras up to isomorphism, and is later
applied to both finite and real base fields [8, 20].

We generalize Petersson’s methodology to $n$-dimensional algebras that are
principal Albert isotopes of $C_{n}$-algebras. This approach is novel in this
generality and again starts by distinguishing between regular and singular
algebras. Petersson relates regular algebras (i.e., algebras where the left
multiplication by some element, and the right multiplication by some element,
are invertible) to unital algebras using a slight generalisation of
Kaplanski’s well-known trick, and introduces the unital heart of an algebra.
He then gives necessary and sufficient criteria for any two algebras of
dimension two with the same unital heart to be isomorphic, without using
structure constants.

Let ${\rm Alg}(V)$ be the set of non-unital algebra structures on an
$n$-dimensional vector space $V$. If $A\in{\rm Alg}(V)$ is regular then the
orbit of $A$ under $G_{0}={\rm Gl}(V)\times{\rm Gl}(V)$ contains a unital
algebra. If this algebra is associative or an octonion algebra, then it is
unique up to isomorphism (Theorem 1.4). We call it the _unital heart_ of $A$.
While this does not allow us any insights into regular algebras with unital
algebras which are different from the above, Theorem 1.4 at least partially
generalizes Petersson’s Unital Heart Proposition and permits a first rough
“classification” of regular algebras by distinguishing between those with
different associative unital hearts, or with different octonion algebras as
their unital heart. We proceed to give necessary and sufficient criteria for
two algebras to be isomorphic, who have the same commutative associative
unital heart (Theorem 1.5).

In the next step we fix a Galois $C_{n}$-algebra $K\in{\rm Alg}(V)$ and
investigate the $n$-dimensional regular algebras whose unital heart is $K$.
Petersson’s approach generalizes well to regular algebras of any dimension
$n$, whose unital heart is a cyclic field extension $K/F$, so we will focus on
these division algebras for most of the paper. Given a cyclic field extension
$K$ of degree $n$ over $F$, we list all the types of division algebras over
the field $F$ whose unital heart is $K$, but only achieve a tight
classification for some subtypes (Theorem 5.2). The situation improves when
$n$ is prime (Theorem 5.3). The Classification Theorem for algebras with a
cubic field extension as unital heart, however, is “tight” in the sense of
Petersson for several cases (Theorem 4.2). On the way, we also classify the
regular algebras with a $C_{3}$-Galois algebra as unital heart (Theorem 6.2).

Thus nonassociative $n$-dimensional division algebras isotopic to cyclic field
extensions are understood up to isomorphism over those base fields, where we
have a full understanding of the cyclic field extensions of degree $n$.

The content of the paper is as follows: After collecting the basic terminology
(Section 1), we describe the bijective linear maps for Galois $C_{n}$-algebras
in Section 2. We then consider regular algebras with a Galois $C_{n}$-algebra
as unital heart (Section 3), and achieve a classification of the three-
dimensional regular algebras with unital heart a cubic Galois field extension
in Section 4. With the same methology, we then achieve a classification of
algebras whose unital heart is a cyclic Galois field extension of degree $n$
in Section 5. In this generality, we are still able to list nonisomorphic
classes of algebras, but are not able to parametrize the algebras to the
extent we were able to in several subcases for the $n=3$ case. We conclude
with a classification of the regular algebras whose unital heart is a
$C_{3}$-Galois algebra in Section 6. This can be generalized to the unital
heart being any Galois $C_{n}$-algebra if desired; we leave this to the
reader.

Most classifications of classes of nonassociative division algebras of
dimension $n$ over some base field $F$ heavily depend on $F$ and on $n$, take
$n=3$ as example: if $F$ is an algebraically closed or a real closed field,
then there are no three-dimensional nonassociative division algebras over $F$.

Over finite fields, three-dimensional nonassociative division algebras exist
and are all isotopic to generalized twisted fields [16, 17]. Indeed, the first
examples of three-dimensional division algebras seem due to Dickson who
constructed three-dimensional unital commutative nonassociative algebras
already in 1905 [3, 4, 5, 6]. Kaplansky then developed their structure theory
in more modern language [10, 11]. Over finite fields of characteristic not
two, any three-dimensional commutative division algebra is a field or one of
these “Dickson” algebras.

There are a range of papers that attempt to classify certain subfamilies of
nonassociative algebras of smaller dimension $n$, often over $\mathbb{C}$,
$\mathbb{R}$ or finite fields, and often employing structural constants:
unital associative algebras over algebraically closed fields of dimension less
than five are classified up to isomorphism in [7]; five-dimensional unital
associative algebras over an algebraically closed field of characteristic not
two are classified up to isomorphism in [15], both approaches use structural
constants. For a classification of two-dimensional nonassociative algebras
over any base field employing structure constants, cf. [1]. While there are
good arguments for preferring structure constants, eg. see [12, 13], we will
avoid them in our classification, as we believe the structure of the algebras
becomes more visible this way.

Singular (i.e., not regular) algebras can be partly investigated by a direct
approach, again following the one used by Petersson. Partial results can be
obtained but will not be treated here, as we wanted to focus mostly on
division algebras.

Our exhibition closely follows [19].

##  1\. Singular and regular algebras and the unital heart

###  1.1. Isotopes and Kaplanski’s trick

Let $F$ be a field. Following [19, Section 1], we denote the set of non-unital
algebra structures on an $F$-vector space $V$ by ${\rm Alg}(V)$. Given
$A\in{\rm Alg}(V)$, we write $xAy$ for the product of $x,y\in V$ in the
algebra $A$ instead of simply juxtaposition, when it is not ad hoc clear which
algebra multiplication is considered. If $A$ is a unital associative algebra,
then we denote the group of invertible elements in $A$ by $A^{\times}$.

An algebra $A\not=0$ is called a _division algebra_ if for any $a\in A$,
$a\not=0$, the left and right multiplication with $a$, $L_{a}:A\rightarrow A$,
$L_{a}(x)=ax$, and $R_{a}:A\rightarrow A$, $R_{a}(x)=xa$, are bijective. If it
is not clear which algebra structure on $V$ is considered, we denote the left
and right multiplication with $a\in V$ by $L_{A}(a)$ and $R_{A}(a)$. If $A$
has finite dimension over $F$, then $A$ is a division algebra if and only if
$A$ has no zero divisors [22, pp. 15, 16].

For $f,g,h\in{\rm Gl}(V)$ define the algebra $A^{(f,g,h)}$, called an isotope
of $A$, as $V$ together with the new multiplication

$xA^{(f,g,h)}y=h(f(x)Ag(y))$

for $x,y\in V$. Two algebras $A,A^{\prime}\in{\rm Alg}(V)$ are called
isotopic, if $xAy=h(f(x)A^{\prime}g(y))$ for all $x,y\in V.$ If $f=g=h^{-1}$
then $A\cong A^{\prime}$. We call $A^{(f,g)}=A^{(f,g,id)}$ a _principal Albert
isotope_ of $A$.

The group $G_{0}={\rm Gl}(V)\times{\rm Gl}(V)$ acts on ${\rm Alg}(V)$
exponentially from the right according to $xA^{(f,g)}y=f(x)Ag(y)$. We know
that $(A^{(f,g)})^{op}=(A^{op})^{(f,g)}$. If $A$ is a division algebra then
$A^{(f,g,h)}$ is a division algebra. (Note that these definitions also make
sense when $f$ and $g$ are not invertible.)

An algebra $A\in{\rm Alg}(V)$ is called _left regular_ , if there exists $u\in
A$ such that $L_{A}(u)\in{\rm Gl}(V)$, otherwise it is _left singular_ , and
_right regular_ if there exists $u\in A$ such that $R_{A}(u)\in{\rm Gl}(V)$,
otherwise it is _right singular_. If $A$ is both left and right regular, it is
called _regular_ , otherwise _singular_. Clearly, $A$ is left regular if and
only if $A^{op}$ is right regular. It also follows easily that, if $A\in{\rm
Alg}(V)$ is regular, so are the isotopes $A^{(f,g)}$ for all $(f,g)\in G_{0}$.
Note that every division algebra is a regular algebra, and that the (principal
Albert) isotope of any division algebra again is a division algebra.

######  Lemma 1.1.

[19, Lemma 1.3] For $A,B\in{\rm Alg}(V)$, $f,g\in{\rm Gl}(V)$, every
isomorphism $\varphi:A\rightarrow B$ is also an isomorphism
$\varphi:A^{(f,g)}\rightarrow B^{(f^{\prime},g^{\prime})}$ where
$f^{\prime}=\varphi f\varphi^{-1}$, and $g^{\prime}=\varphi g\varphi^{-1}$.

Division algebras are principal Albert isotopes of unital division algebras,
more generally so are regular algebras [19, 1.5]:

######  Proposition 1.2.

(Kaplansky’s Trick) Let $A\in{\rm Alg}(V)$ be a regular algebra and choose
$u,v\in V$ such that $f=R_{A}(v)$ and $g=L_{A}(u)$ are bijective. Then
$B=A^{(f^{-1},g^{-1})}$ is a unital algebra over $F$ with unit element $e=uAv$
and $A=B^{(f,g)}$.

######  Lemma 1.3.

Let $K\in{\rm Alg}(V)$ be a unital commutative associative algebra and
$f,g\in{\rm Gl}(V)$. Write $L=L_{K}$ and $R=R_{K}$. Then the following are
equivalent:
(i) $K^{(f,g)}$ is a unital algebra over $F$.
(ii) $K^{(f,g)}$ is a _Jordan isotope_ of $K$, i.e. there exists $w\in
K^{\times}$ such that $K^{(f,g)}=K^{(w)}$, with $K^{(w)}$ defined via
$xK^{(w)}y=xwy$ for all $x,y\in V$.
(iii) $K^{(f,g)}\cong K$.
Moreover, in this case $w$ satisfies $w^{-1}=1_{K^{(f,g)}}$ and is uniquely
determined, and there are $u,v\in K$ such that $w=uv$, $f=L(u)$, $g=R(v)$.
A map $\varphi:K^{(f,g)}\rightarrow K$ is an isomorphism if and only if
$\varphi=\sigma L(w)$ for some $\sigma\in{\rm Aut}(K)$.

This is the generalization of [19, Lemma 1.9] which was stated only for $n=2$.
Its proof holds verbatim for any $n$-dimensional unital commutative
associative algebra $K$.

######  Theorem 1.4.

(The Unital Heart) For all $A\in{\rm Alg}(V)$, the following statements are
equivalent:
(i) $A$ is regular.
(ii) There exist a unital algebra $K\in{\rm Alg}(V)$ and $f,g\in{\rm Gl}(V)$
such that $A\cong K^{(f,g)}$.
If $K$ is associative, then $K$ is unique up to isomorphism and is called the
_unital heart_ of $A$. If $K$ is an octonion algebra, then $K$ is unique up to
isomorphism and is called the _unital heart_ of $A$.

This generalizes [19, Proposition 1.10] which was formulated for $n=2$. It
remains an open problem how or if this approach might work for regular
algebras that are Albert isotopes of a unital algebra that is not associative
and not an octonion algebra.

###### Proof.

The proof that (i) and (ii) are equivalent given in [19, Proposition 1.10]
holds verbatim for any $n$-dimensional unital algebra $K$. It remains to show
that $K$ is unique up to isomorphism: Let $K,K^{\prime}\in{\rm Alg}(V)$ be
unital and associative with $K\cong K^{\prime(f^{\prime},g^{\prime})}$ for
some $f^{\prime},g^{\prime}\in{\rm Gl}(V)$. Then there are
$f^{\prime\prime},g^{\prime\prime}\in{\rm Gl}(V)$ such that $K^{\prime}\cong
K^{(f^{\prime\prime},g^{\prime\prime})}$. We thus have two associative unital
algebras $K,K^{\prime}$ that are isotopic. It is well-known that this means
$K\cong K^{\prime}$. Let $K,K^{\prime}\in{\rm Alg}(V)$ be two octonion
algebras with $K\cong K^{\prime(f^{\prime},g^{\prime})}$ for some
$f^{\prime},g^{\prime}\in{\rm Gl}(V)$. Then again there are
$f^{\prime\prime},g^{\prime\prime}\in{\rm Gl}(V)$ such that $K^{\prime}\cong
K^{(f^{\prime\prime},g^{\prime\prime})}$. By [2, Proposition 2], this means
$K\cong K^{\prime}$. ∎

######  Theorem 1.5.

(Isomorphism Criterium) Let $K\in{\rm Alg}(V)$ be a unital commutative
associative algebra and $(f,g)$, $(f^{\prime},g^{\prime})\in G$. Write
$L=L_{K}$ and $R=R_{K}$. Then the following statements are equivalent:
(i) $\varphi:K^{(f,g)}\rightarrow K^{(f^{\prime},g^{\prime})}$ is an
isomorphism.
(ii) There exist $u,v\in K^{\times}$, $\sigma\in{\rm Aut}(K)$ such that

$f^{\prime}=L(v^{-1})\sigma f\sigma^{-1}L(uv),\quad g^{\prime}=L(v^{-1})\sigma
g\sigma^{-1}L(uv),\quad\varphi=L((uv)^{-1})\sigma.$

This generalizes [19, 1.12] to any dimension, the proof is verbatim the same.
Note that the proof of $(ii)\Rightarrow(i)$ in [19, 1.12] needs that $K$ is
associative and that $u\in{\rm Comm}(K)$.

######  Theorem 1.6.

(Isomorphism Criterium for Octonion Algebras) [2, Proposition 4] Let $K\in{\rm
Alg}(V)$ be a quaternion algebra or an octonion algebra and $(f,g)$,
$(f^{\prime},g^{\prime})\in G_{0}$. Write $L=L_{K}$ and $R=R_{K}$. Then the
following statements are equivalent:
(i) $\varphi:K^{(f,g)}\rightarrow K^{(f^{\prime},g^{\prime})}$ is an
isomorphism.
(ii) $\varphi$ is a similitude of the norm $N_{K}$ of $K$ and there are
similitudes $\varphi_{i}$ of $N_{K}$ with
$\varphi(xy)=\varphi_{1}(x)\varphi_{2}(y)$ for all $x,y\in K$ and

$f^{\prime}=\varphi_{1}f\varphi^{-1},\quad
g^{\prime}=\varphi_{2}g\varphi^{-1}$

or equivalently,

$f^{\prime}=R_{\varphi_{2}(1_{K})}^{-1}\varphi f\varphi^{-1},\quad
g^{\prime}=L_{\varphi_{1}(1_{K})}^{-1}\varphi g\varphi^{-1}.$

As a side remark, we observe:

######  Corollary 1.7.

Let $K\in{\rm Alg}(V)$ be a unital associative algebra, such that its norm
$N(x)=\det L(x)$ is a form of degree $n$ permitting composition, and $(f,g)\in
G_{0}$.
(i) The pair

$(\det(f)\,{\rm mod}\,N(K^{\times}),\det(g)\,{\rm mod}\,N(K^{\times}))$

is an invariant of $K^{(f,g)}$ in
$(F^{\times}/N(K^{\times}),F^{\times}/N(K^{\times}))$.
(ii) For all $a,b\in F^{\times}$, we have $K^{(f,g)}\cong K^{(af,bg)}$.

###### Proof.

(i) The proof generalizes the one of [19, Corollary 1.13]: let
$f,g,f^{\prime},g^{\prime}\in{\rm Gl}(V)$. We compare the determinants in
Theorem 1.5 (ii) and obtain

$\det(f^{\prime})=\det(L(v^{-1})\det(\sigma)\det(f)\det(\sigma^{-1})\det(L(uv))=\det(f)\det(L(v^{-1})\det(L(uv))$

and analogously, $\det(g^{\prime})=\det(g)\det(L(v^{-1})\det(L(uv)).$ Using
that $N(x)=\det L(x)$ for $x\in V$ we get
$\det(f^{\prime})=\det(f)N(v^{-1})N(uv)$ and
$\det(g^{\prime})=\det(g)N(v^{-1})N(uv).$ Since $N$ permits composition it
follows that $\det(f^{\prime})=\det(f)N(u)$ and
$\det(g^{\prime})=\det(g)N(u),$ and we obtain the assertion.
(ii) This generalizes [19, Corollary 1.14]. The proof is identical to the one
of [19, Corollary 1.14] and uses Theorem 1.5. ∎

######  Corollary 1.8.

Let $K\in{\rm Alg}(V)$ be a quaternion or octonion algebra and $(f,g)\in
G_{0}$. Then

$K^{(f,g)}\cong K^{(af,bg)}$

for all $a,b\in F^{\times}$.

###### Proof.

Let $a,b\in F^{\times}$, $c=ab$, then $\varphi_{1}(x)=ax$,
$\varphi_{2}(x)=bx$, $\varphi_{1}(x)=cx$ are norm similarities of $N$ and
$\varphi(xy)=\varphi_{1}(x)\varphi_{2}(y)$ for all $x,y\in K$. We have
$f^{\prime}(x)=\varphi_{1}(f(\varphi^{-1}(x))))=\varphi_{1}(f(c^{-1}x))=ac^{-1}f(x)=b^{-1}x$
and
$g^{\prime}(x)=\varphi_{2}(g(\varphi^{-1}(x))))=\varphi_{2}(g(c^{-1}x))=bc^{-1}f(x)=a^{-1}x$.
By Theorem 1.6, then $K^{(f,g)}\cong K^{(b^{-1}f,a^{-1}g)}$. Choosing $a^{-1}$
and $b^{-1}$ instead of $b,a$ in the calculation yields the assertion. ∎

We can now “sort” nonassociative regular algebras over $F$ of dimension $n$ by
their unital heart. This breaks up the classification of regular algebras with
(well-defined) unital hearts into two steps: (i) the classification of the
possible unital hearts of dimension $n$, i.e. of the unital associative
algebras $K\in{\rm Alg}(V)$ of dimension $n$, and of the octonion algebras
$K\in{\rm Alg}(V)$, and then (ii), the classification of the algebras with the
same unital heart. Unlike in the two-dimensional case, however, even the class
of unital associative algebras is now in general formidable, and does not only
contain commutative associative algebras anymore like for $n=2$.

######  Remark 1.9.

The restriction to unital commutative associative algebras, respectively, to
octonion algebras as unital hearts is clearly limiting the scope of the
classification we can achieve with this method. Even in small degrees, and
over an algebraically closed field (of characteristic not two), there are many
commutative associative algebras: among the 59 unital associative algebras of
dimension 5 listed up to isomorphism in [15], 19 are commutative. For a
complete lists of low dimensional complex associative nonunital algebras see
[21].

##  2\. Invertible linear maps

###  2.1. Invertible linear maps for Galois $C_{n}$-algebras

A commutative associative algebra $K\in{\rm Alg}(V)$ of dimension $n$ is
called étale, if $K\cong F_{1}\times\dots\times F_{r}$ for some finite
separable field extensions $F_{1},\dots,F_{r}$ of a field $F$. Let $G$ be a
group of $F$-automorphisms of $K$. An étale algebra $K$ over $F$ of dimension
$n$ is a Galois $G$-algebra over $F$, if the order of $G$ equals $n$, and if
$K$ can be endowed with an action by $G$ such that ${\rm Fix}(G)=\\{x\in
K\,|\,g(x)=x\text{ for all }g\in G\\}=F$ [14, (18.B.), p. 287]. A Galois
$G$-algebra structure on a field $K$ exists if and only if the field extension
$K/F$ is Galois with Galois group isomorphic to $G$ [14, (18.16), p. 288].

Let $V$ be an $n$-dimensional $F$-vector space. Let $K\in{\rm Alg}(V)$ be a
Galois $C_{n}$-algebra, i.e. a cyclic Galois field extension of $F$ of degree
$n$ with Galois group ${\rm Gal}(K/F)=\langle\tau\rangle$ or an
$n$-dimensional étale algebra over $F$ with a cyclic automorphism group
$\langle\tau\rangle$ of order $n$, with norm $N$ and trace $T$.

Take the cyclic algebra $A_{0}=(K/F,\tau,1)=K[t;\sigma]/(t^{n}-1)$ of degree
$n$ with reduced norm $N_{A_{0}/F}:A_{0}\rightarrow F$, $N_{A_{0}/F}(x)={\rm
det}(L_{x}).$ Consider $A_{0}$ as a left $K$-module with basis
$\\{1,t,\ldots,t^{n-1}\\}$. Write $L(x)$ for the matrix of left multiplication
by $x=x_{0}+x_{1}t+\cdots+x_{n-1}t^{n-1},$ $x_{i}\in K$ in $A_{0}$ with
respect to the basis $\\{1,t,\ldots,t^{n-1}\\}:$

$L(x)=\left(\begin{array}[]{cccccc}x_{0}&\tau(x_{n-1})&...&\tau^{n-1}(x_{1})\\\
x_{1}&\tau(x_{0})&...&\tau^{n-1}(x_{2})\\\
x_{2}&\tau(x_{1})&...&\tau^{n-1}(x_{3})\\\ ...&...&...&...\\\
x_{m-2}&\tau(x_{n-3})&...&\tau^{n-1}(x_{n-1})\\\
x_{m-1}&\tau(x_{n-2})&...&\tau^{n-1}(x_{0})\\\ \end{array}\right).$

######  Theorem 2.1.

(cf. [19, Proposition 2.2] for $n=2$) Let $K\in{\rm Alg}(V)$ be a Galois
$C_{n}$-algebra with nontrivial automorphism $\tau$. Let $V$ be an
$n$-dimensional $F$-vector space. Then the map $(K/F,\tau,1)\to{\rm
End}_{F}(V)$,

$(x_{0},\dots,x_{n-1})\mapsto
L(x_{0})+L(x_{1})\tau+L(x_{2})\tau^{2}+\dots+L(x_{n-1})\tau^{n-1},$

is an isomorphism of central simple algebras. In particular, we can write

${\rm End}_{F}(V)=L(K)\oplus L(K)\tau\oplus\dots\oplus L(K)\tau^{n-1}$

as a direct sum of $F$-subvector spaces and

$\det(L(x_{0})+L(x_{1})\sigma+L(x_{2})\tau^{2}+\dots+L(x_{n-1})\tau^{n-1})=N_{A_{0}/F}(x_{0}+x_{1}t+\cdots+x_{n-1}t^{n-1}).$

###### Proof.

The map $(x_{0},\dots,x_{n-1})\mapsto
L(x_{0})+L(x_{1})\tau+L(x_{2})\tau^{2}+\dots+L(x_{n-1})\tau^{n-1}$ is a
homomorphism. This is an immediate consequence of the structure of the matrix
$L(x)$ which defines the multiplication in $A_{0}$ and the fact that we have
$\sigma L(x)=L(\sigma(x))\sigma$ for all $x\in K$ and
$\sigma\in\langle\tau\rangle$. A homomorphism between two central simple
algebras of the same dimension is an isomorphism of central simple algebras.

The second statement is now obvious. The third follows from the fact that
norms of central simple algebras are preserved by isomorphisms, but was also
independently previously observed in [23]. ∎

######  Proposition 2.2.

Assume we are in the situation of Theorem 2.1. Suppose that $n=rs$ for some
positive integers $s\geq 2$, $s\not=n$. Let $E={\rm Fix}(\tau^{s})$ and let
$W$ be an $r$-dimensional $E$-vector space. Then the map

$(x_{0},x_{s},\dots,x_{s(r-1)})\mapsto
L(x_{0})+L(x_{s})\tau^{s}+\dots+L(x_{s(r-1)})\tau^{s(r-1)}$

induced by restricting the map $(K/F,\tau,1)\to{\rm End}_{F}(V)$ in Theorem
2.1 to $B_{0}=(K/E,\tau^{s},1)$, is an isomorphism of the central simple
algebras $(K/E,\tau^{s},1)$ and ${\rm End}_{E}(W)$ over $E$, where both
algebras are now viewed as $F$-algebras. In particular, we can write

${\rm End}_{E}(W)=L(K)\oplus L(K)\tau^{s}\oplus\dots\oplus L(K)\tau^{s(r-1)}$

as a direct sum of $E$-subvector spaces and

$\det(L(x_{0})+L(x_{s})\tau^{s}+\dots+L(x_{s(r-1)})\tau^{s(r-1)})=N_{B_{0}/E}(x_{0}+x_{1}t^{s}+\cdots+x_{s(r-1)}t^{s(r-1)}).$

###### Proof.

The fact that the map is an isomorphism of the central simple algebras
$(K/E,\tau^{s},1)$ and ${\rm End}_{E}(W)$ of degree $r$ over $E$ is shown
analogously as in Theorem 2.1. The algebra $(K/E,\tau^{s},1)$ is an
$F$-subalgebra of $(K/F,\tau,1)$ and so is ${\rm End}_{E}(W)$ of ${\rm
End}_{F}(V)$. Restricting an isomorphisms $(K/F,\tau,1)\to{\rm End}_{F}(V)$ to
the subalgebra $(K/E,\tau^{s},1)$ yields exactly the isomorphisms
$(K/E,\tau^{s},1)\to{\rm End}_{E}(W)$, where both algebras are viewed as
$F$-algebras. ∎

######  Corollary 2.3.

Assume we are in the situation of Theorem 2.1. Then every $f\in{\rm
End}_{F}(V)$ can be uniquely written in the form

(1) $f=L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1}$

for some $(y_{0},y_{1},\dots,y_{n-1})\in K^{n}$ and lies in ${\rm Gl}(V)$ if
and only if $N_{A_{0}}(y_{0}+y_{1}t+\dots+y_{n-1}t^{n-1})\not=0$.
If $n=rs$ for $s\geq 2$, $s\not=n$, $E={\rm Fix}(\tau^{s})$, then every
$f\in{\rm End}_{F}(V)$ that can be written in the form

$f=L(y_{0})+L(y_{s})\tau^{s}+\dots+L(y_{s(r-1)})\tau^{s(r-1)}$

for some $(y_{0},y_{s},\dots,y_{s(r-1)})\in K^{r}$, is an isomorphism of
$(K/E,\tau^{s},1)$ and ${\rm End}_{E}(W)$ over $E$ when restricted to
$(K/E,\tau^{s},1)$, where $W$ is an $r$-dimensional $E$-vector space.

###  2.2. Invertible linear maps for any abelian Galois field extension

We can push this approach a bit further: Let $V$ be an $n$-dimensional
$F$-vector space. Let $K\in{\rm Alg}(V)$ be a Galois field extension of $F$ of
degree $n$ with abelian Galois group $G={\rm
Gal}(K/F)=\\{\tau_{1},\dots,\tau_{n}\\}$.

Take the crossed product algebra $A_{0}=(K/F,G,\Phi)$ of degree $n$ with the
2-cocycle $\Phi$ defined by $\Phi(\sigma,\tau)=1$ for all $\sigma,\tau\in G$
and reduced norm $N_{A_{0}/F}:A_{0}\rightarrow F$, $N_{A_{0}/F}(x)={\rm
det}(L_{x}).$ Consider $A_{0}$ as a left $K$-vector space with basis
$\\{1,u_{\tau_{1}},\ldots,u_{\tau_{n-1}}\\}$. Write $L(x)$ for the matrix of
left multiplication by
$x=x_{0}+x_{1}u_{\tau_{1}}+\cdots+x_{n-1}u_{\tau_{n-1}},$ $x_{i}\in K$ in
$A_{0}$. A straightforward calculation shows that

$L(x)=\left(\begin{array}[]{cccccc}x_{0}&\tau_{1}(x_{n-1})&...&\tau_{n-1}(x_{1})\\\
x_{1}&\tau_{1}(x_{0})&...&\tau_{n-1}(x_{2})\\\
x_{2}&\tau_{1}(x_{1})&...&\tau_{n-1}(x_{3})\\\ ...&...&...&...\\\
x_{n-2}&\tau_{1}(x_{n-3})&...&\tau_{n-1}(x_{n-1})\\\
x_{n-1}&\tau_{1}(x_{n-2})&...&\tau_{n-1}(x_{0})\\\ \end{array}\right).$

######  Theorem 2.4.

Let $V$ be an $n$-dimensional $F$-vector space. Let $K\in{\rm Alg}(V)$ be a
Galois field extension of $F$ of degree $n$ with abelian Galois group
$G=\\{\tau_{1},\dots,\tau_{n}\\}$. The map $(K/F,G,\Phi)\to{\rm End}_{F}(V),$

$(x_{0},\dots,x_{n-1})\mapsto
L(x_{0})+L(x_{1})\tau_{1}+L(x_{2})\tau^{2}+\dots+L(x_{n-1})\tau_{n-1},$

is an isomorphism of central simple algebras. In particular, we can write

${\rm End}_{F}(V)=L(K)\oplus L(K)\tau_{1}\oplus\dots\oplus L(K)\tau_{n-1}$

as a direct sum of $F$-subvector spaces and

$\det(L(x_{0})+L(x_{1})\tau_{1}+L(x_{2})\tau_{2}+\dots+L(x_{n-1})\tau_{n-1})=N_{A_{0}/F}(x_{0}+x_{1}u_{\tau_{1}}+\cdots+x_{n-1}u_{\tau_{n-1}}).$

###### Proof.

The map $(x_{0},\dots,x_{n-1})\mapsto
L(x_{0})+L(x_{1})\tau_{1}+L(x_{2})\tau_{2}+\dots+L(x_{n-1})\tau_{n-1}$ is a
homomorphism. This is an immediate consequence of the structure of the matrix
$L(x)$ which defines the multiplication in $A_{0}$ and the fact that we have
$\tau_{i}L(x)=L(\tau_{i}(x))\tau_{i}$ for all $x\in K$ and suitable integers
$i$. A homomorphism between two central simple algebras of the same dimension
is an isomorphism of central simple algebras.

The second statement is now obvious. The third follows again from the fact
that norms of central simple algebras are preserved by isomorphisms. ∎

######  Corollary 2.5.

In the situation of Theorem 2.4, every $f\in{\rm End}_{F}(V)$ can be uniquely
written in the form

(2) $f=L(y_{0})+L(y_{1})\tau+L(y_{2})\tau_{2}+\dots+L(y_{n-1})\tau_{n-1}$

for some $(y_{0},y_{1},\dots,y_{n-1})\in K^{n}$ and lies in ${\rm Gl}(V)$ if
and only if
$N_{A_{0}}(y_{0}+y_{1}u_{\tau_{1}}+\cdots+y_{n-1}u_{\tau_{n-1}})\not=0$.

We often write $N_{A_{0}/F}(y)=N_{A_{0}/F}(y_{0}+y_{1}t+\dots+y_{n-1}t^{n-1})$
from now on.

Analogous results holds for Galois $G$-algebras.

######  Example 2.6.

Let $char(F)\not=2$, $\dim_{F}V=4$ and $K\in{\rm Alg}(V)$ be a biquadratic
field extension $K=F(\sqrt{u},\sqrt{v})$ with ${\rm
Gal}(K/F)=\\{1,\sigma,\tau,\sigma\tau\\},$ where $\sigma(\sqrt{u})=\sqrt{u},$
$\sigma(\sqrt{v})=-\sqrt{v},$ $\tau(\sqrt{u})=-\sqrt{u}$,
$\tau(\sqrt{v})=\sqrt{v}.$ Let $F_{0}=F(\sqrt{v})={\rm Fix}(\tau)$. Take the
crossed product algebra $A_{0}=(K/F,G,\Phi)$ of degree $n$ with the 2-cocycle
$\Phi$ defined by $\Phi(\tau_{i},\tau_{j})=1$ for all $\tau_{i}\in G$. Here,

$L(x)=\left(\begin{array}[]{cccc}x_{0}&\sigma(x_{3})&\tau(x_{2})&\sigma\tau(x_{1})\\\
x_{1}&\sigma(x_{0})&\tau(x_{3})&\sigma\tau(x_{2})\\\
x_{2}&\sigma(x_{1})&\tau(x_{0})&\sigma\tau(x_{3})\\\
x_{3}&\sigma(x_{2})&\tau(x_{1})&\sigma\tau(x_{0})\\\ \end{array}\right).$

Let $K$ have norm $N$ and trace $T$, then every $f\in{\rm End}_{F}(V)$ can be
uniquely written in the form

$f=L(y_{0})+L(y_{1})\sigma+L(y_{2})\tau+L(y_{3})\sigma\tau$

for some $(y_{0},y_{1},y_{2},y_{3})\in K^{4}$ and lies in ${\rm Gl}(V)$ if and
only if

$\displaystyle
N_{A_{0}}(y_{0}+y_{1}u_{\sigma}+y_{2}u_{\tau}+y_{3}u_{\sigma\tau})$
$\displaystyle=N(y_{0})-N(y_{1})+N(y_{2})-N(y_{3})$
$\displaystyle+T(y_{0}\sigma(y_{1})\tau(y_{1})\sigma\tau(y_{2}))-T(y_{0}\sigma(y_{0})\tau(y_{1})\sigma\tau(y_{3}))$
$\displaystyle+T(y_{0}\sigma(y_{2})\tau(y_{3})\sigma\tau(y_{3}))+T(y_{1}\sigma(y_{2})\tau(y_{2})\sigma\tau(y_{3}))$
$\displaystyle-
T_{F_{0}/F}(y_{0}\tau(y_{0})\sigma(y_{2})\sigma\tau(y_{2}))+T_{F_{0}/F}(y_{1}\tau(y_{1})\sigma(y_{3})\sigma\tau(y_{3}))\not=0.$

##  3\. Regular algebras with a Galois $C_{n}$-algebra as unital heart

Let $V$ be an $n$-dimensional $F$-vector space. Let $K\in{\rm Alg}(V)$ be a
Galois $C_{n}$-algebra with cyclic automorphism group generated by $\tau$. Put
$N=N_{K/F}$. Define the set of idempotents ${\rm Id}(K)=\\{d\in
K\,|\,d^{2}=d\\}$, and note that ${\rm Id}(K)=\\{0,1\\}$ if $K$ is a field.

######  Lemma 3.1.

Suppose that $y_{i},y_{i}^{\prime}\in K$ satisfy
$N_{A/F}(y_{0}+y_{1}t+\cdots+y_{n-1}t^{n-1})\not=0$ and
$N_{A/F}(y_{0}^{\prime}+y^{\prime}_{1}t+\cdots+y^{\prime}_{n-1}t^{n-1})\not=0$.
(i) We have

$K^{(L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1},g)}\cong
K^{(L(y_{0}^{\prime})+L(y^{\prime})\tau+L(y^{\prime}_{2})\tau^{2}+\dots+L(y^{\prime}_{n-1})\tau^{n-1},g^{\prime})}$

if and only if there are $u,v\in K^{\times}$ and $\sigma\in{\rm Aut}_{F}(K)$,
such that

$g^{\prime}=L(u^{-1})\sigma g\sigma^{-1}L(uv)$

and

$y_{0}^{\prime}=u\sigma(y_{0}),\quad
y_{i}^{\prime}=\tau^{i}(u)\tau^{i}(v)v^{-1}\sigma(y_{i})$

for all $i\in\\{1,\dots,n-1\\}$.
(ii) Let $y_{0}=d,y_{0}^{\prime}=d^{\prime}\in{\rm Id}(K)$. If the statements
in (i) hold then $d^{\prime}=\sigma(d)$ and $d=1$ forces $d^{\prime}=u=1$.
(iii) For all $i\in\\{0,\dots,n-1\\}$, $y_{i}$ is invertible if and only if
$y_{i}^{\prime}$ is, and nonzero if and only so is $y_{i}^{\prime}$.

###### Proof.

(i) By Theorem 1.5,

$K^{(L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1},g)}\cong
K^{(L(y_{0}^{\prime})+L(y^{\prime})\tau+L(y^{\prime}_{2})\tau^{2}+\dots+L(y^{\prime}_{n-1})\tau^{n-1},g^{\prime})}$

if and only if there are $u,v\in K^{\times}$, $\sigma\in{\rm Aut}_{F}(K)$,
such that $g^{\prime}=L(u^{-1})\sigma g\sigma^{-1}L(uv)$ and

$L(y_{0}^{\prime})+L(y_{1}^{\prime})\tau+L(y^{\prime}_{2})\tau^{2}+\dots+L(y^{\prime}_{n-1})\tau^{n-1}$
$=L(v^{-1})\sigma(L((y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1})\sigma^{-1}L(uv)$
$=L(v^{-1})(L(\sigma(y_{0}))+L(\sigma(y_{1}))\tau+L(\sigma(y_{2}))\tau^{2}+\dots+L(\sigma(y_{n-1}))\tau^{n-1})L(uv)$
$=L(u\sigma(y_{0}))+L(\tau(u)\tau(v)v^{-1}\sigma(y_{1}))\tau+L(\tau^{2}(u)\tau^{2}(v)v^{-1}\sigma(y_{2}))\tau^{2}+$
$\dots+L(\tau^{n-1}(u)\tau^{n-1}(v)v^{-1}\sigma(y_{n-1}))\tau^{n-1}.$

Compare components as in (2) to get

$y_{0}^{\prime}=u\sigma(y_{0}),$
$y_{1}^{\prime}=\tau(u)\tau(v)v^{-1}\sigma(y_{1}),\quad
y_{2}^{\prime}=\tau^{2}(u)\tau^{2}(v)v^{-1}\sigma(y_{2}),\dots,y_{n-1}^{\prime}=\tau^{n-1}(u)\tau^{n-1}(v)v^{-1}\sigma(y_{n-1})).$

(ii) Let $y_{0}=d$, $y_{0}^{\prime}=d^{\prime}\in{\rm Id}(K)$. If the
statements are fulfilled, we have $u\sigma(d)=d^{\prime}=d^{\prime
2}=u^{2}\sigma(d)^{2}=u^{2}\sigma(d)$. If $d\not=0$ then this implies $u=1$
thus $d^{\prime}=\sigma(d).$ If $d=0$ then $d^{\prime}=\sigma(d)$ is trivially
true, since $u\not=0$. The statement is now obvious.
(iii) is trivial. ∎

######  Lemma 3.2.

Let $K\in{\rm Alg}(V)$ be a Galois $C_{n}$-algebra with ${\rm
Aut}_{F}(K)=\langle\tau\rangle$, norm $N$ and trace $T$. Suppose that
$f=L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1}\in{\rm
Gl}(V)$. Let $u,v\in K^{\times}$ and $\sigma\in{\rm Aut}_{F}(K)$, and put

$y_{0}^{\prime}=u\sigma(y_{0}),\quad
y_{i}^{\prime}=\tau^{i}(u)\tau^{i}(v)v^{-1}\sigma(y_{i})$

for all $i\in\\{1,\dots,n-1\\}$. Define
$f^{\prime}=L(y_{0}^{\prime})+L(y^{\prime})\tau+L(y^{\prime}_{2})\tau^{2}+\dots+L(y^{\prime}_{n-1})$.
Then $f^{\prime}\in{\rm Gl}(V)$.

###### Proof.

To better demonstrate the idea of the proof, we start with $n=3$ and $n=4$. If
$n=3$ then
$N_{A_{0}/F}(y_{0}+y_{1}t+y_{2}t^{2})=N(y_{0})+N(y_{1})+N(y_{2})-T(d\tau(y_{1})\tau^{2}(y_{2}))$.
Now $N_{K}(y_{i}^{\prime})=N(u)N(y_{i})$ for $i=1,2$ and
$N(y_{0}^{\prime})=N(u\sigma(y_{0}))=N(u)N(y_{0})$, moreover

$d^{\prime}\tau(y^{\prime}_{1})\tau^{2}(y^{\prime}_{2})=u\sigma(y_{0})\tau^{2}(u)\tau^{2}(v)\tau(v^{-1})\tau(\sigma(y_{1}))\tau(u)\tau(v)\tau^{2}(v^{-1})\tau^{2}(\sigma(y_{2}))$
$=N(u)\sigma(y_{0})\tau(\sigma(y_{1}))\tau^{2}(\sigma(y_{2}))=N_{K}(u)\sigma(y_{0}\tau(y_{1})\tau^{2}(y_{2})).$

Therefore

$T(N(u)\sigma(y_{0}\tau(y_{1})\tau^{2}(y_{2})))=N(u)T(d\tau(y_{1})\tau^{2}(y_{2})).$

Thus

$N_{A_{0}/F}(y_{0}^{\prime}+y^{\prime}_{1}t+y^{\prime}_{2}t^{2})=N(u)N_{A_{0}/F}(y_{0}+y_{1}t+y_{2}t^{2}),$

which means that $f^{\prime}$ is invertible if and only if so is $f$.

Let $n=4$ and $F_{0}={\rm Fix}(\tau^{2})=\\{x\in K\,|\,\tau^{2}(x)=x\\}$
(i.e., the intermediate field of $K$ and $F$ fixed by $\tau^{2}$ in case $K/F$
is a Galois field extension). Then

$\displaystyle
N_{A_{0}/F}(y_{0}^{\prime}+y^{\prime}_{1}t+y^{\prime}_{2}t^{2}+y^{\prime}_{3}t^{3})$
$\displaystyle=N(y_{0}^{\prime})-N(y^{\prime}_{1})+N(y^{\prime}_{2})$
$\displaystyle-N(y^{\prime}_{3})+T(d^{\prime}\tau(y^{\prime}_{1})\tau^{2}(y^{\prime}_{1})\tau^{3}(y^{\prime}_{2}))$
$\displaystyle-T(y_{0}^{\prime}\tau(y_{0}^{\prime})\tau^{2}(y^{\prime}_{1})\tau^{3}(y^{\prime}_{3}))$
(3)
$\displaystyle+T(y_{0}^{\prime}\tau(y^{\prime}_{2})\tau^{2}(y^{\prime}_{3})\tau^{3}(y^{\prime}_{3}))$
$\displaystyle+T(y^{\prime}_{1}\tau(y^{\prime}_{2})\tau^{2}(y^{\prime}_{2})\tau^{3}(y^{\prime}_{3}))$
$\displaystyle-
T_{F_{0}/F}(d^{\prime}\tau^{2}(d^{\prime})\tau(y^{\prime}_{2})\tau^{3}(y^{\prime}_{2}))$
$\displaystyle+T_{F_{0}/F}(y^{\prime}_{1}\tau^{2}(y^{\prime}_{1})\tau(y^{\prime}_{3})\tau^{3}(y^{\prime}_{3}))$

(cf. [23], which holds analogously when $K$ is a Galois $C_{4}$-algebra, as it
is the determinant of the same matrix, but now with entries in the
$C_{4}$-algebra $K$). An easy calculation shows that again
$N_{A_{0}/F}(y_{0}^{\prime}+y^{\prime}_{1}t+y^{\prime}_{2}t^{2}+y^{\prime}_{3}t^{3})=N(u)N_{A_{0}/F}(y_{0}+y_{1}t+y_{2}t^{2}+y_{3}t^{3})$.

In the general case we proceed as follows: the matrix of left multiplication
by $y_{0}+y_{1}t+\cdots+y_{n-1}t^{n-1}$ in $A_{0}$ with respect to the basis
$\\{1,t,\ldots,t^{n-1}\\}$ is given by

$L(y)=\left(\begin{array}[]{cccccc}y_{0}&\tau(y_{n-1})&...&\tau^{n-1}(y_{1})\\\
y_{1}&\tau(y_{0})&...&\tau^{n-1}(y_{2})\\\
y_{2}&\tau(y_{1})&...&\tau^{n-1}(y_{3})\\\ ...&...&...&...\\\
y_{n-2}&\tau(y_{n-3})&...&\tau^{n-1}(y_{n-1})\\\
y_{n-1}&\tau(y_{n-2})&...&\tau^{n-1}(y_{0})\\\ \end{array}\right)$

and satisfies
$\det(L(y))=\det(L(y_{0},\dots,y_{n-1}))=\det(L(\tau(y_{0}),\dots,\tau(y_{n-1})))$
for all $y_{i}\in K$ ([18, Proposition 1] holds for any base field and indeed
for any $C_{n}$-algebra). More precisely, $\det(L(y))$ is a sum of terms of
the type

$\rho_{0}(y_{0})\rho_{1}(x_{1})\cdots\rho_{n-1}(y_{n-1})$

with $n_{i}\in\mathbb{Z}$ and the $\rho_{i}$ are either 0 or
$\rho_{i}(x)=\tau^{r_{1}}(x)\cdots\tau^{r_{s}}(x)$, where the
$r_{j}\in\mathbb{Z}_{m}$ are all distinct. That means $\det(L(y))$ is a sum of
elements that look like $f_{s_{0},s_{1},\dots,s_{n-1}}(y_{0},\dots,y_{n-1}),$
where all the $f_{s_{0},s_{1},\dots,s_{n-1}}(y_{0},\dots,y_{n-1})$ are
polynomial functions in $y_{i}$, $s_{i}\in\mathbb{Z}$, and where
$f_{j}(y_{0},\dots,y_{n-1})\in F$ for all choices of $y_{i}\in K$. This is
proved by using clever matrix manipulations analogously as in [18, Corollary
1].

In particular, we see that for each such term
$\rho_{0}(y_{0})\rho_{1}(y_{1})\cdots\rho_{n-1}(y_{n-1})$, we have

$\rho_{0}(y^{\prime}_{0})\rho_{1}(y^{\prime}_{1})\cdots\rho_{n-1}(y^{\prime}_{n-1})=N(u)\rho_{0}(y_{0})\rho_{1}(y_{1})\cdots\rho_{n-1}(y_{n-1}).$

That means
$N_{A_{0}/F}(y^{\prime})=\det(L(y^{\prime}))=N(u)\det(L(y^{\prime}))=N(u)N_{A_{0}/F}(y)$
which proves the assertion. ∎

The first step towards a classification of algebras that have a cyclic Galois
field extension of degree $n$ as unital heart is the following observation
that lists the different types of $(f,g)\in G$:

######  Theorem 3.3.

Let $K\in{\rm Alg}(V)$ be a cyclic Galois field extension of degree $n$. Let
$(f,g)\in G$. Then there exist $(f^{\prime},g^{\prime})\in G_{0}$ such that
$K^{(f,g)}=K^{(f^{\prime},g^{\prime})}$ and $f^{\prime}$ satisfies exactly one
of the following conditions:

  1. (1)

$f^{\prime}=id+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1}$ for
some $y_{i}\in K^{\times}$ such that
$N_{A_{0}/F}(1+y_{1}t+\dots+y_{n-1}t^{n-1})\not=0$.

  2. (2)

$f^{\prime}=id+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1}$ and
there is a nonempty set $\\{i_{1},\dots,i_{s}\\}\subset\\{1,\dots,n-1\\}$ of
less than $n-1$ elements, such that all $y_{i_{1}},\dots,y_{i_{s}}$ equal 0,
all $y_{i}\in\\{1,\dots,n-1\\}\setminus\\{i_{1},\dots,i_{s}\\}$ are nonzero,
and $N_{A_{0}/F}(y)\not=0$. Every different subset $\\{i_{1},\dots,i_{s}\\}$
is a different subtype.

  3. (3)

$f^{\prime}=id+L(0)\tau+L(0)\tau^{2}+\dots+L(0)\tau^{n-1}=1$.

  4. (4)

$f^{\prime}=L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1}$ for some
$y_{i}\in K^{\times}$ such that $N_{A_{0}/F}(y)\not=0$.

  5. (5)

$f^{\prime}=L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1}$ for some
nonempty set $\\{i_{1},\dots,i_{s}\\}\subset\\{1,\dots,n-1\\}$ of less than
$n-1$ elements, such that all $y_{i_{1}},\dots,y_{i_{s}}$ are zero, all
$y_{i}\in\\{1,\dots,n-1\\}\setminus\\{i_{1},\dots,i_{s}\\}$ are nonzero, and
$N_{A_{0}/F}(y)\not=0$. Every different subset $\\{i_{1},\dots,i_{s}\\}$ is a
different subtype.

Two different subtypes in (2), respectively in (5), yield nonisomorphic Albert
isotopes $K^{(f,g)}$.

###### Proof.

By Theorem 2.1,
$f=L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1}$ for some
$y=y_{0}+y_{1}t+\dots+y_{n-1}t^{n-1}\in A_{0}$ such that
$N_{A_{0}/F}(y)\not=0$. Moreover, if $y_{0}\in K^{\times}$ then there is
$y_{0}^{-1}\in K^{\times}$ such that $1=y_{0}^{-1}y_{0}$. Applying the
isomorphism criterium with $u=y_{0}^{-1}$, $\sigma=id$ we may assume
$y_{0}=1$. This leads us to two cases/types of isotopes, those with $y_{0}=1$
and those with $y_{0}=0$. The map $f^{\prime}$ we construct in the following
is an isomophism by Lemma 3.2.
Case $y_{0}=1$:
(a) $y_{i}\in K^{\times}$ for all $i\in\\{1,\dots,n-1\\}$,
(b) There is a nonempty set
$N=\\{i_{1},\dots,i_{s}\\}\subset\\{1,\dots,n-1\\}$ of less than $n-1$
elements, such that $y_{i_{1}},\dots,y_{i_{s}}$ are zero and the elements
$y_{1},\dots,y_{n-1}$ that do not lie in $N$ are nonzero.
(c) For all $i\geq 1$, $y_{i}=0$.
Case $y_{0}=0$:
(a) $y_{i}\in K^{\times}$ for all $i\in\\{1,\dots,n-1\\}$,
(b) There is a nonempty set
$N=\\{i_{1},\dots,i_{s}\\}\subset\\{1,\dots,n-1\\}$ of less than $n-1$
elements, such that $y_{i_{1}},\dots,y_{i_{s}}$ are zero and the elements that
do not lie in $N$ are nonzero (apart from $y_{0}$). ∎

This lists grows when $K\in{\rm Alg}(V)$ is, more generally, a Galois
$C_{n}$-algebra.

In the following we will abuse the notion of a “partition” by allowing the
sets $I$, $N$ and $S$ involved to be empty:

######  Proposition 3.4.

Let $K\in{\rm Alg}(V)$ be a Galois $C_{n}$-algebra with automorphism group
generated by $\tau$. Let $(f,g)\in G$. Then there exist
$(f^{\prime},g^{\prime})\in G_{0}$ such that
$K^{(f,g)}=K^{(f^{\prime},g^{\prime})}$ and $f^{\prime}$ satisfies exactly one
of the following conditions:

  1. (1)

$f^{\prime}=id+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1}$ and
there are pairwise disjoint sets $I,N,S\subset\\{1,\dots,n-1\\}$ with $I\cup
N\cup S=\\{1,\dots,n-1\\}$, such that all $y_{i}\in N$ equal 0, all
$y_{i}\not\in N$ are nonzero, all $y_{i}\in S$ lie in $K\setminus K^{\times}$,
all $y_{i}\in I$ lie in $K^{\times}$, and $N_{A_{0}/F}(y)\not=0$.

  2. (2)

There is $y_{0}\in K\setminus K^{\times}$, $y_{0}\not=0$, such that
$f^{\prime}=L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1}$
and there are pairwise disjoint sets $I,N,S\subset\\{1,\dots,n-1\\}$ with
$I\cup N\cup S=\\{1,\dots,n-1\\}$, such that all $y_{i}\in N$ equal 0, all
$y_{i}\not\in N$ are nonzero, all $y_{i}\in S$ lie in $K\setminus K^{\times}$,
all $y_{i}\in I$ lie in $K^{\times}$, and $N_{A_{0}/F}(y)\not=0$.

  3. (3)

$f^{\prime}=L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1}$ and
there are pairwise disjoint sets $I,N,S\subset\\{1,\dots,n-1\\}$, with $I\cup
N\cup S=\\{1,\dots,n-1\\}$, such that $N$ has less than $n-1$ elements, all
$y_{i}\in N$ equal 0, all $y_{i}\not\in N$ are nonzero, all $y_{i}\in S$ lie
in $K\setminus K^{\times}$, all $y_{i}\in I$ lie in $K^{\times}$, and
$N_{A_{0}/F}(1+y_{1}t+\dots+y_{n-1}t^{n-1})\not=0$. Every different partition
$I,N,S$ is a different subtype.

The types listed in (1) to (5) in Theorem 3.3 are contained in the above list
under the types in (1) and (3) when $S=\emptyset$. Two different subtypes of
the same type yield nonisomorphic Albert isotopes $K^{(f,g)}$.

###### Proof.

By Theorem 2.1, we have

$f=L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1}$

for some $y=y_{0}+y_{1}t+\dots+y_{n-1}t^{n-1}\in A_{0}$ such that
$N_{A_{0}/F}(y)\not=0$. If $y_{0}\in K^{\times}$ then applying the isomorphism
criterium with $u=y_{0}^{-1}$, $\sigma=id$ we may assume $y_{0}=1$. The
remaining cases are $y_{0}=0$ and $y_{0}\in K\setminus K^{\times}$.

We now look at the different cases for the other $y_{i}$’s. In each case, we
distinguish the different possible “partitions” of $\\{1,\dots,n-1\\}$,
allowing the sets $N$, $I$ $S$ to be empty to cover all possible cases. In
order to obtain a bijective $f^{\prime}$, we require that
$N_{A_{0}/F}(y_{0}+y_{1}t+\dots+y_{n-1}t^{n-1})\not=0$ in each case. ∎

######  Corollary 3.5.

Let $K\in{\rm Alg}(V)$ be a cubic Galois field extension. Let $(f,g)\in
G_{0}$. Then there exist $(f^{\prime},g^{\prime})\in G_{0}$ such that
$K^{(f,g)}=K^{(f^{\prime},g^{\prime})}$ and $f^{\prime}$ satisfies exactly one
of the following conditions:

  1. (1)

$f^{\prime}=id+L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $y_{i}\in K^{\times}$
with $N_{K}(y_{1})+N_{K}(y_{2})-T_{K}(\tau(y_{1})\tau^{2}(y_{2}))\not=-1$.

  2. (2)

$f^{\prime}=id+L(y_{2})\tau^{2}$ for some $y_{2}\in K^{\times}$ with
$N_{K}(y_{2})\not=-1$.

  3. (3)

$f^{\prime}=id+L(y_{1})\tau$ for some $y_{1}\in K^{\times}$ with
$N_{K}(y_{1})\not=-1$.

  4. (4)

$f^{\prime}=id$.

  5. (5)

$f^{\prime}=L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $y_{i}\in K^{\times}$ with
$N_{K}(y_{2})\not=-N_{K}(y_{1})$.

  6. (6)

$f^{\prime}=L(y_{2})\tau^{2}$ for some $y_{2}\in K^{\times}$ with
$N_{K}(y_{2})\not=0$.

  7. (7)

$f^{\prime}=L(y_{1})\tau$ for $y_{1}\in K^{\times}$ with $N_{K}(y_{1})\not=0$.

###### Proof.

Since
$N_{A_{0}/F}(1+y_{1}t+y_{2}t^{2})=N_{K}(y_{0})+N_{K}(y_{1})+N_{K}(y_{2})-T_{K}(y_{0}\tau(y_{1})\tau^{2}(y_{2}))$
we get the following cases:
(1) $f^{\prime}=id+L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $y_{i}\in
K^{\times}$ and
$N_{A_{0}/F}(1+y_{1}t+y_{2}t^{2})=1+N_{K}(y_{1})+N_{K}(y_{2})-T_{K}(\tau(y_{1})\tau^{2}(y_{2}))\not=0$.
(2) $f^{\prime}=id+L(y_{2})\tau^{2}$ such that $y_{2}\in K^{\times}$ with
$N_{A_{0}/F}(1+y_{2}t^{2})=1+N_{K}(y_{2})\not=0$, which is equivalent to
$N_{K}(y_{2})\not=-1$.
(3) $f^{\prime}=id+L(y_{1})\tau$ such that $y_{1}\in K^{\times}$ with
$N_{A_{0}/F}(1+y_{1}t)=1+N_{K}(y_{1})\not=0$, which is equivalent to
$N_{K}(y_{1})\not=-1$.
(4) This is $f^{\prime}=id+L(0)\tau+L(0)\tau^{2}$.
(5) $f^{\prime}=L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $y_{i}\in K^{\times}$
such that $N_{A_{0}/F}(y_{1}t+y_{2}t^{2})=N_{K}(y_{1})+N_{K}(y_{2})\not=0$,
which is equivalent to $N_{K}(y_{2})\not=-N_{K}(y_{1})$.
(6) $f^{\prime}=L(0)\tau+L(y_{2})\tau^{2}$ for $y_{2}\in K^{\times}$ with
$N_{A_{0}/F}(y_{2}t^{2})=N_{K}(y_{2})\not=0$,
(7) $f^{\prime}=L(y_{1})\tau+L(0)\tau^{2}$ for $y_{1}\in K^{\times}$ with
$N_{A_{0}/F}(y_{1}t)=N_{K}(y_{1})\not=0$. ∎

######  Corollary 3.6.

Let $K\in{\rm Alg}(V)$ be a $C_{3}$-Galois algebra and write
$N_{A_{0}}(y)=N_{A_{0}/F}(1+y_{1}t+y_{2}t^{2})$. Let $(f,g)\in G_{0}$. Then
there exist $(f^{\prime},g^{\prime})\in G_{0}$ such that
$K^{(f,g)}=K^{(f^{\prime},g^{\prime})}$ and $f^{\prime}$ satisfies exactly one
of the following conditions: (1) to (7) from Corollary 3.5,

  1. (8)

$f^{\prime}=id+L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $0\not=y_{i}\in
K\setminus K^{\times}$ with $N_{A_{0}}(y)\not=0$.

  2. (9)

$f^{\prime}=id+L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $0\not=y_{1}\in
K\setminus K^{\times}$, $y_{2}\in K^{\times}$ with $N_{A_{0}}(y)\not=0$.

  3. (10)

$f^{\prime}=id+L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $y_{1}\in K^{\times}$,
$0\not=y_{2}\in K\setminus K^{\times}$ with $N_{A_{0}}(y)\not=0$.

  4. (11)

$f^{\prime}=L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $0\not=y_{i}\in K\setminus
K^{\times}$ with $N_{A_{0}}(y)\not=0$.

  5. (12)

$f^{\prime}=L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $0\not=y_{1}\in K\setminus
K^{\times}$, $y_{2}\in K^{\times}$ with $N_{A_{0}}(y)\not=0$.

  6. (13)

$f^{\prime}=L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $y_{1}\in K^{\times}$,
$0\not=y_{2}\in K\setminus K^{\times}$ with $N_{A_{0}}(y)\not=0$.

  7. (14)

$f^{\prime}=L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $0\not=y_{0}\in
K\setminus K^{\times}$, $y_{1},y_{2}\in K^{\times}$ with $N_{A_{0}}(y)\not=0$.

  8. (15)

$f^{\prime}=L(y_{0})+L(y_{2})\tau^{2}$ for some $y_{2}\in K^{\times}$ with
$N_{A_{0}}(y_{0}+y_{2}t^{2})\not=0$.

  9. (16)

$f^{\prime}=L(y_{0})+L(y_{1})\tau$ for some $y_{1}\in K^{\times}$ with
$N_{A_{0}}(y)\not=0$.

  10. (17)

$f^{\prime}=L(y_{0})+L(y_{2})\tau^{2}$ for some $0\not=y_{2}\in K\setminus
K^{\times}$ with $N_{A_{0}}(y)\not=0$.

  11. (18)

$f^{\prime}=L(y_{0})+L(y_{1})\tau$ for some $0\not=y_{1}\in K\setminus
K^{\times}$ with $N_{A_{0}}(y)\not=0$.

  12. (19)

$f^{\prime}=L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $0\not=y_{0}\in
K\setminus K^{\times}$, $0\not=y_{i}\in K\setminus K^{\times}$ with
$N_{A_{0}}(y)\not=0$.

  13. (20)

$f^{\prime}=L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $0\not=y_{0}\in
K\setminus K^{\times}$, $0\not=y_{1}\in K\setminus K^{\times}$, $y_{2}\in
K^{\times}$ with $N_{A_{0}}(y)\not=0$.

  14. (21)

$f^{\prime}=L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $0\not=y_{0}\in
K\setminus K^{\times}$, $y_{1}\in K^{\times}$, $0\not=y_{2}\in K\setminus
K^{\times}$ with $N_{A_{0}}(y)\not=0$.

###### Proof.

Cases (1) to (7) are proved analogously as when $K$ is a field. Additionally,
we now also have to distinguish when $0\not=y_{i}\in K\setminus K^{\times}$
which leads to the other new cases: We note that $N_{A_{0}}(y)\not=0$ is
equivalent to $y\in A_{0}$ being invertible, so that the condition that
$N_{A_{0}/F}(1+y_{1}t+y_{2}t^{2})\not=0$ excludes certain cases. ∎

Define $S(K)=\\{x\in V\,|\,N(x)=1\\}$. By Hilbert’s Theorem 90, we know

$S(K)=\\{\frac{\tau(v)}{v}\,|\,v\in K^{\times}\\}.$

Choose a set of representatives $M$ of $K^{\times}/S(K)$ containing 1.

######  Lemma 3.7.

Let $K\in{\rm Alg}(V)$ be a $C_{3}$-Galois algebra. Let
$y_{i},y_{i}^{\prime}\in M$ and $u,v\in K^{\times}$, $\sigma\in{\rm
Aut}_{F}(K)$, such that
$y_{i}^{\prime}=\tau^{i}(u)\tau^{i}(v)v^{-1}\sigma(y_{i})$ for some
$i\in\\{1,\dots,n-1\\}$ and $N(u)=1$. Then $y_{i}=y_{i}^{\prime}$.

###### Proof.

Since $y_{i}^{\prime}=\tau^{i}(u)\tau^{i}(v)v^{-1}\sigma(y_{i})$ we have
$N(y_{i}^{\prime})=N(y_{i})$, thus $s:=y_{i}^{\prime}y_{i}^{-1}\in S(K)$. The
equation $y_{i}=sy_{i}^{\prime}$ means that $y_{i},y_{i}^{\prime}\in M$ are
congruent modulo $S(K)$, thus $y_{i}=y_{i}^{\prime}$. ∎

We will now use the set $M$ to attempt a classification of our algebras.

##  4\. A classification of division algebras with a cubic Galois field
extension as the unital heart

Let $A$ be a regular algebra over $F$ of dimension 3 with a unital heart
isomorphic to a cubic Galois field extension $K$ with ${\rm
Gal}(K/F)=\langle\tau\rangle$. Then $A$ is a division algebra and we know that
$A\cong K^{(f,g)}$ for some $(f,g)\in G_{0}$ by Theorem 1.4. By Corollary 3.5
we are left with the following cases:
(1) $f=id+L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $y_{i}\in K^{\times}$ such
that $N_{K}(y_{1})+N_{K}(y_{2})-T_{K}(\tau(y_{1})\tau^{2}(y_{2}))\not=-1$:
Then there exists an element $y_{1}^{\prime}\in M$, such that
$y_{1}^{\prime}=y_{1}\,{\rm mod}\,S(K)$, i.e. there exists $v\in K^{\times}$
such that $y_{1}^{\prime}=\tau(v)v^{-1}y_{1}$. Set $\sigma=id$, $u=1$ and
$g^{\prime}=gL(v)$ in Lemma 3.1. Then for $y_{1}^{\prime}=\tau(v)v^{-1}y_{1},$
$y_{2}^{\prime}=\tau^{2}(v)v^{-1}y_{2}$ we have

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(1+L(y_{1}^{\prime})\tau+L(y^{\prime}_{2})\tau^{2},g^{\prime})}.$

(2) $f=id+L(y_{2})\tau^{2}$ for some $y_{2}\in K^{\times}$ with
$N_{K}(y_{2})\not=-1$: Then there exists an element $y_{2}^{\prime}\in M$,
such that $y_{2}^{\prime}=y_{2}\,{\rm mod}\,S(K)$, i.e. there exists $v\in
K^{\times}$ such that $y_{2}^{\prime}=\tau^{2}(v)v^{-1}y_{2}$.

Set $\sigma=id$ and $u=1$ in Lemma 3.1. Then choose $g^{\prime}=gL(v)$ and for
$y_{2}^{\prime}=\tau^{2}(v)v^{-1}y_{2}$ we have

$K^{(1+L(y_{2})\tau^{2},g)}\cong
K^{(1+L(y^{\prime}_{2})\tau^{2},g^{\prime})}.$

(3) $f=id+L(y_{1})\tau$ for some $y_{1}\in K^{\times}$ with
$N_{K}(y_{1})\not=-1$: Again there exists an element $y_{1}^{\prime}\in M$,
such that $N_{K}(y_{1})=N_{K}(y_{1}^{\prime})$, and there exists $v\in
K^{\times}$ such that $y_{1}^{\prime}=\tau(v)v^{-1}y_{1}$. Set $\sigma=id$,
$u=1$ and $g^{\prime}=gL(v)$ in Lemma 3.1. Then for
$y_{1}^{\prime}=\tau(v)v^{-1}y_{1}$ we have

$K^{(1+L(y_{1})\tau,g)}\cong K^{(1+L(y^{\prime}_{1})\tau,g^{\prime})}.$

(4) $f=id$: Then $g=L(x_{0})+L(x_{1})\tau+L(x_{2})\tau^{2}$ for $x_{i}\in K$
with $N_{A_{0}}(x_{0}+x_{1}t+x_{2}t^{2})\not=0$. We have two cases to
consider: $x_{0}=0$, then choose $v=1$; $x_{0}\in K^{\times}$, then choose
$v=x_{0}^{-1}$. Apply Lemma 3.1 for $u=1$, $v$, $\sigma=id$ to see that for
$g^{\prime}=gL(v)$ we obtain $K^{(1,g)}\cong K^{(1,g^{\prime})}$. Now
$g^{\prime}=L(e)+L(x_{1})\tau+L(x_{2})\tau^{2}$ for $e,x_{i}\in K$ with
$N_{A_{0}}(e+x_{1}t+x_{2}t^{2})\not=0$ and $e\in\\{0,1\\}$.
(4.i) If $e=0$ then $g^{\prime}=L(x_{1})\tau+L(x_{2})\tau^{2}$ for $x_{i}\in
K$ with $N_{A_{0}}(x_{1}t+x_{2}t^{2})\not=0$.
(4.ii) If $e=1$ then $g^{\prime}=1+L(x_{1})\tau+L(x_{2})\tau^{2}$ for
$x_{i}\in K$ with $N_{A_{0}}(1+x_{1}t+x_{2}t^{2})\not=0$.
(5) $f=L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $y_{i}\in K^{\times}$ with
$N_{K}(y_{2})\not=-N_{K}(y_{1})$: There exists an element $y_{1}^{\prime}\in
M$, such that $N_{K}(y_{1})=N_{K}(y_{1}^{\prime})$ and $v\in K^{\times}$ such
that $y_{1}^{\prime}=\tau(v)v^{-1}y_{1}$. Set $\sigma=id$ and $u=1$ in Lemma
3.1. Then for $g^{\prime}=gL(v)$ and for
$y_{2}^{\prime}=\tau^{2}(v_{1})v_{1}^{-1}y_{2}$ we get

$K^{(L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(L(y_{1}^{\prime})\tau+L(y^{\prime}_{2})\tau^{2},g^{\prime})}$

by Lemma 3.1.
(6) $f=L(y_{2})\tau^{2}$ for some $y_{2}\in K^{\times}$ with
$N_{K}(y_{2})\not=0$: Again there exists an element $y_{2}^{\prime}\in M$,
such that $N_{K}(y_{2})=N_{K}(y_{2}^{\prime})$ and $v\in K^{\times}$ such that
$y_{2}^{\prime}=\tau^{2}(v)v^{-1}y_{2}$. Set $\sigma=id$ and $u=1$ in Lemma
3.1. Then for $g^{\prime}=gL(v)$ we obtain

$K^{(L(y_{2})\tau^{2},g)}\cong K^{(L(y^{\prime}_{2})\tau^{2},g^{\prime})}.$

(7) $f=L(y_{1})\tau$ for $y_{1}\in K^{\times}$ with $N_{K}(y_{1})\not=0$:
Analogously as above, we use Lemma 3.1 for $\sigma=id$, $u=1$, and $v$ with
$y_{1}^{\prime}=\tau(v)v^{-1}y_{1}$. We get $K^{(L(y_{1})\tau,g)}$ for
$y_{1}\in M$, such that $N_{K}(y_{1})\not=0$ and $g\in{\rm Gl(V)}$. We obtain:

######  Theorem 4.1.

(Tight Enumeration Theorem for cubic unital hearts) Let $A$ be a division
algebra of dimension three with unital heart isomorphic to a cubic Galois
field extension, ${\rm Gal}(K/F)=\langle\tau\rangle$. Then $A$ is isomorphic
to precisely one of the following types (in the following, $g\in{\rm Gl(V)}$):

  1. (1)

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2},g)}$ for $y_{1}\in M$, $y_{2}\in
K^{\times}$ with
$N_{K}(y_{1})+N_{K}(y_{2})-T_{K}(\tau(y_{1})\tau^{2}(y_{2}))\not=-1$.

  2. (2)

$K^{(1+L(y_{2})\tau^{2},g)}$ for $y_{2}\in M$, such that
$N_{K}(y_{2})\not=-1$.

  3. (3)

$K^{(1+L(y_{1})\tau,g)}$ for $y_{1}\in M$, such that $N_{K}(y_{1})\not=-1$.

  4. (4)

$K^{(1,g)}$ with $g=L(x_{1})\tau+L(x_{2})\tau^{2}$ for $x_{i}\in K$ with
$N_{A_{0}}(x_{1}t+x_{2}t^{2})\not=0$.

  5. (5)

$K^{(1,g)}$ with $g=1+L(x_{1})\tau+L(x_{2})\tau^{2}$ for $x_{i}\in K$ with
$N_{A_{0}}(1+x_{1}t+x_{2}t^{2})\not=0$.

  6. (6)

$K^{(L(y_{1})\tau+L(y_{2})\tau^{2},g)}$ for $y_{1}\in M$, $y_{2}\in
K^{\times}$, such that $N_{K}(y_{2})\not=-N_{K}(y_{1})$.

  7. (7)

$K^{(L(y_{2})\tau^{2},g)}$ for $y_{2}\in M$ with $N_{K}(y_{2})\not=0$.

  8. (8)

$K^{(L(y_{1})\tau,g)}$ for $y_{1}\in M$ with $N_{K}(y_{1})\not=0$.

###### Proof.

By Lemma 3.1, two algebras $A$ and $B$ that belong to different types (1),
(2), (3) and (6), (7), (8) cannot be isomorphic. The same holds for algebras
that lie in either (4) or (5), again these are not isomorphic to any of the
types listed in (1), (2), (3) or (6), (7), (8). Since
$(A^{(f,g)})^{op}\cong(A^{op})^{(g,f)}$ we can apply Lemma 3.1 to the opposite
algebras as well which shows us that algebras of different type (4) and (5)
also are not isomorphic. ∎

To complete the classification of three-dimensional algebras which have a
cubic Galois field extension as unital heart, we now check when two algebras
of the same type are isomorphic:

######  Theorem 4.2.

(Classification Theorem for algebras with a cubic Galois field extension as
unital heart) Let $K/F$ be a cubic Galois field extension, ${\rm
Gal}(K/F)=\langle\tau\rangle$.

  1. (1)

For $y_{1},z_{1}\in M$, $y_{2},z_{2}\in K^{\times}$ with
$N_{K}(y_{1})+N_{K}(y_{2})-T_{K}(\tau(y_{1})\tau^{2}(y_{2}))\not=-1$,
$N_{K}(z_{1})+N_{K}(z_{2})-T_{K}(\tau(z_{1})\tau^{2}(z_{2}))\not=-1$ and
$g,g^{\prime}\in{\rm Gl(V)}$, we have

$K^{(L(d)+L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(L(d^{\prime})+L(y^{\prime})\tau+L(y^{\prime}_{2})\tau^{2},g^{\prime})}$

if and only if $z_{1}=y_{1}$ and exactly on of the following holds:
(i) $z_{2}=y_{2}$ and there exists $a\in F^{\times}$ such that
$g^{\prime}=ag,$,
(ii) $z_{2}=\tau^{2}(y_{1}^{-1})y_{1}\tau(y_{2})$ and there exists $a\in
F^{\times}$ such that $g^{\prime}=a\tau g\tau^{-1}L(y_{1})^{-1}$,
(iii) $z_{2}=\tau^{2}(y_{1}^{-1})y_{1}\tau^{2}(y_{2})$ and there exists $a\in
F^{\times}$ such that $g^{\prime}=a\tau^{2}g\tau^{-2}L(y_{1})^{-1}.$

  2. (2)

For $y_{2},z_{2}\in M$, such that $N_{K}(y_{2})\not=-1$, $N_{K}(z_{2})\not=-1$
and $g,g^{\prime}\in{\rm Gl(V)}$, we have

$K^{(1+L(y_{2})\tau^{2},g)}\cong K^{(1+L(z_{2})\tau^{2},g^{\prime})}$

if and only if $z_{2}=y_{2}$ and exactly on of the following holds:
(i) there exists $a\in F^{\times}$ such that $g^{\prime}=ag,$
(ii) there exists $a\in F^{\times}$ such that
$g^{\prime}=a\tau^{2}g\tau^{-2}L(y_{2})^{-1}$,
(iii) there exists $v\in K^{\times}$ such that $g^{\prime}=a\tau
g\tau^{-1}L(v).$

  3. (3)

For $y_{1},z_{1}\in M$, such that $N_{K}(y_{1})\not=-1$, $N_{K}(z_{1})\not=-1$
and $g,g^{\prime}\in{\rm Gl(V)}$, we have

$K^{(1+L(y_{1})\tau,g)}\cong K^{(1+L(z_{1})\tau,g)}$

if and only if $z_{1}=y_{1}$ and exactly on of the following holds:
(i) there exists $a\in F^{\times}$ such that $g^{\prime}=ag,$
(ii) there exists $a\in F^{\times}$ such that
$g^{\prime}=a\tau^{i}g\tau^{-i}L(y_{1})^{-1}$ for some $i\in\\{1,2\\}.$

  4. (4)

For $x_{i},w_{i}\in K$ with $N_{A_{0}}(x_{1}t+x_{2}t^{2})\not=0$,
$N_{A_{0}}(w_{1}t+w_{2}t^{2})\not=0$, we have

$K^{(1,L(x_{1})\tau+L(x_{2})\tau^{2})}\cong
K^{(1,L(w_{1})\tau+L(w_{2})\tau^{2})}$

if and only if there exist $v\in K^{\times}$ and
$\sigma\in\\{id,\tau,\tau^{2}\\}$, such that $w_{1}=\sigma(x_{1})\tau(v)$ and
$w_{2}=\sigma(x_{2})\tau^{2}(v)$.

  5. (5)

For $x_{i},w_{i}\in K$ with $N_{A_{0}}(1+x_{1}t+x_{2}t^{2})\not=0$,
$N_{A_{0}}(1+w_{1}t+w_{2}t^{2})\not=0$ we have

$K^{(1,1+L(x_{1})\tau+L(x_{2})\tau^{2})}\cong
K^{(1,1+L(w_{1})\tau+L(w_{2})\tau^{2})}$

if and only if there exists $\sigma\in\\{id,\tau,\tau^{2}\\}$, such that
$w_{1}=\sigma(x_{1})$ and $w_{2}=\sigma(x_{2})$.

  6. (6)

For $y_{1},z_{1}\in M$, $y_{2},z_{2}\in K^{\times}$, such that
$N_{K}(y_{2})\not=-N_{K}(y_{1})$, $N_{K}(z_{2})\not=-N_{K}(z_{1})$ and
$g,g^{\prime}\in{\rm Gl(V)}$, we have

$K^{(L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(L(z_{1})\tau+L(z_{2})\tau^{2},g)}$

if and only if there are $u,v\in K^{\times}$ and
$\sigma\in\\{id,\tau,\tau^{2}\\}$, such that $g^{\prime}=L(u^{-1})\sigma
g\sigma^{-1}L(uv)$ and

$z_{1}=\tau(u)\tau(v)v^{-1}\sigma(y_{1}),\quad
z_{2}=\tau^{2}(u)\tau^{2}(v)v^{-1}\sigma(y_{2}).$

This implies that $N(z_{1})=N(u)N(y_{1})$ and $N(z_{2})=N(u)N(y_{2})$ is
necessary for the above isomorphism to hold.
If $N(u)=1$ then $z_{1}=y_{1}$ and $y_{1}=\tau(u)\tau(v)v^{-1}\sigma(y_{1})$.

  7. (7)

For $y_{2},z_{2}\in M$ with $N_{K}(y_{2})\not=0$, $N_{K}(z_{2})\not=0$ and
$g,g^{\prime}\in{\rm Gl(V)}$, we have

$K^{(L(y_{2})\tau^{2},g)}\cong K^{(L(z_{2})\tau^{2},g)}$

if and only if there are $u,v\in K^{\times}$ and
$\sigma\in\\{id,\tau,\tau^{2}\\}$ such that $g^{\prime}=L(u^{-1})\sigma
g\sigma^{-1}L(uv)$ and

$z_{2}=\tau^{2}(u)\tau^{2}(v)v^{-1}\sigma(y_{2}).$

This implies that $N(z_{2})=N(u)N(y_{2})$ is necessary for the above
isomorphism to hold. If $N(u)=1$ we get $z_{2}=y_{2}$. If $N(u)=1$ we get
$z_{2}=y_{2}$ and $y_{2}=\tau^{2}(u)\tau^{2}(v)v^{-1}\sigma(y_{2})$.

  8. (8)

For $y_{1},z_{1}\in M$ with $N_{K}(y_{1})\not=0$, $N_{K}(z_{1})\not=0$ and
$g,g^{\prime}\in{\rm Gl(V)}$, we have

$K^{(L(y_{1})\tau,g)}\cong K^{(L(z_{1})\tau,g)}$

if and only if there are $u,v\in K^{\times}$ and
$\sigma\in\\{id,\tau,\tau^{2}\\}$, such that $g^{\prime}=L(u^{-1})\sigma
g\sigma^{-1}L(uv)$ and

$z_{1}=\tau(u)\tau(v)v^{-1}\sigma(y_{1}).$

This implies that $N(z_{1})=N(u)N(y_{1})$ is necessary for the above
isomorphism to hold. If $N(u)=1$ then $z_{1}=y_{1}$ and
$y_{1}=\tau(u)\tau(v)v^{-1}\sigma(y_{1})$.

###### Proof.

By Lemma 3.1, two algebras mentioned in any of the cases of Theorem 3.5 are
isomorphic if and only if there are $u,v\in K^{\times}$ and $\sigma\in{\rm
Aut}_{F}(K)$, such that the criteria of Lemma 3.1 are satisfied. We call them
_critical relations_ following [19]. Here, $\sigma\in\\{id,\tau,\tau^{2}\\}$.
By Lemma 3.7, if $z_{i},y_{i}\in M$ and
$z_{i}=\tau^{i}(v)\tau^{i}(v)v^{-1}\sigma(y_{i})$ then $z_{i}=y_{i}$. We go
through the cases of Theorem 3.5:
(1) Since we have $u=1$ here, the critical relations are
$z_{1}=\tau(v)v^{-1}\sigma(y_{1})$, $z_{2}=\tau^{2}(v)v^{-1}\sigma(y_{2})$ and
$g^{\prime}=\sigma g\sigma^{-1}L(v)$, which implies $z_{1}=y_{1}$ since
$z_{1},y_{1}\in M$.

Assume $\sigma=id$. Using $z_{1}=y_{1}$ and $y_{1}=\tau(v)v^{-1}\sigma(y_{1})$
this yields $\tau(v)v^{-1}=1$, that means $\tau(v)=v$, so $v=a\in F^{\times}$.
Therefore the critical relations here become $z_{1}=y_{1}$, $z_{2}=y_{2}$ and
there exists $a\in F^{\times}$ such that $g^{\prime}=ag$.

Assume $\sigma=\tau$, i.e. $y_{1}=\tau(v)v^{-1}\tau(y_{1})$ which is the same
as $\tau(vy_{1})=vy_{1}$, i.e. $vy_{1}\in F^{\times}$. Now the critical
relations become: there is $a\in F^{\times}$, $v=ay_{1}^{-1}$ and
$z_{2}=\tau^{2}(y_{1}^{-1})y_{1}\tau(y_{2})$, $g^{\prime}=a\tau
g\tau^{-1}L(y_{1})^{-1}$.

Assume $\sigma=\tau^{2}$, i.e. the critical relations are
$y_{1}=\tau^{2}(v)v^{-1}\tau^{2}(y_{1})$, which is the same as
$\tau^{2}(vy_{1})=vy_{1}$, i.e. $a=vy_{1}\in F^{\times}$,
$z_{2}=\tau^{2}(v)v^{-1}\tau^{2}(y_{2})$ and
$g^{\prime}=\tau^{2}g\tau^{-2}L(v)$. Hence there is $a\in F^{\times}$,
$v=ay_{1}^{-1}$ and $y_{1}=z_{1}$,
$z_{2}=\tau^{2}(y_{1}^{-1})y_{1}\tau^{2}(y_{2})$,
$g^{\prime}=a\tau^{2}g\tau^{-2}L(y_{1})^{-1}$.
(2) Since $u=1$, here the critical relations are
$z_{2}=\tau^{2}(v)v^{-1}\sigma(y_{2})$ and $g^{\prime}=\sigma
g\sigma^{-1}L(v)$, which implies $z_{2}=y_{2}$ by the definition of $M$.

Assume $\sigma=id$. Using $z_{2}=y_{2}$, the equation
$z_{2}=\tau^{2}(v)v^{-1}\sigma(y_{2})$ yields $\tau^{2}(v)v^{-1}=1$, that
means $\tau^{2}(v)=v$, so $v=a\in F^{\times}$. Therefore the critical
relations are $z_{2}=y_{2}$ and there exists $a\in F^{\times}$ such that $v=a$
and $g^{\prime}=ag$.

Assume $\sigma=\tau$. Then the critical relations are
$y_{2}=\tau^{2}(v)v^{-1}\tau(y_{2})$ and $g^{\prime}=\tau g\tau^{-1}L(v)$,

Assume $\sigma=\tau^{2}$. Then the critical relations are
$y_{2}=\tau^{2}(v)v^{-1}\tau^{2}(y_{2})$ and
$g^{\prime}=\tau^{2}g\tau^{-2}L(v)$. The first equation is the same as
$\tau^{2}(vy_{2})=vy_{2}$, i.e. $a=vy_{2}\in F^{\times}$ and
$g^{\prime}=\tau^{2}g\tau^{-2}L(v)$. Hence there is $a\in F^{\times}$, such
that $v=ay_{2}^{-1}$ and $z_{2}=\tau^{2}(y_{2}^{-1})y_{2}\tau^{2}(y_{2})$,
$g^{\prime}=a\tau^{2}g\tau^{-2}L(y_{2})^{-1}$.
(3) Here the critical relations imply that $z_{1}=\tau(v)v^{-1}\sigma(y_{1})$
and $g^{\prime}=\sigma g\sigma^{-1}L(v)$, hence $z_{1}=y_{1}$ by the
definition of $M$.

Assume $\sigma=id$. This yields $\tau(v)v^{-1}=1$, that means $\tau(v)=v$, so
$v=a\in F^{\times}$. Therefore the critical relations are $z_{1}=y_{1}$ and
there exists $a\in F^{\times}$ such that $g^{\prime}=ag$.

Assume $\sigma=\tau$, i.e. $y_{1}=\tau(v)v^{-1}\tau(y_{1})$. Then there is
$a\in F^{\times}$, $v=ay_{1}^{-1}$ and $g^{\prime}=a\tau
g\tau^{-1}L(y_{1})^{-1}$.

Assume $\sigma=\tau^{2}$, i.e. the critical relations are
$y_{1}=\tau^{2}(v)v^{-1}\tau^{2}(y_{1})$, which is the same as
$\tau(vy_{1})=vy_{1}$, i.e. $a=vy_{1}\in F^{\times}$, and
$g^{\prime}=\tau^{2}g\tau^{-2}L(v)$. Therefore the critical relations are
$z_{1}=y_{1}$ and $g^{\prime}=a\tau^{2}g\tau^{-2}L(y_{1})^{-1}$.
(4) $K^{(1,g)}$ with $g=L(x_{1})\tau+L(x_{2})\tau^{2}$ for $x_{i}\in K$ with
$N_{A_{0}}(x_{1}t+x_{2}t^{2})\not=0$. Here, the critical relations reduce to
$u=1$ and

$L(w_{1})\tau+L(w_{2})\tau^{2}=\sigma(L(x_{1})\tau+L(x_{2})\tau^{2})\sigma^{-1}L(v)$
$=(L(\sigma(x_{1}))\tau L(v)+L(\sigma(x_{2}))\tau^{2}L(v)$
$=L(\sigma(x_{1})\tau(v))\tau+L(\sigma(x_{2})\tau^{2}(v))\tau^{2}.$

By Corollary 2.3, this is equivalent to $w_{1}=\sigma(x_{1})\tau(v)$ and
$w_{2}=\sigma(x_{2})\tau^{2}(v).$
(5) Here, the critical relations reduce to $u=1$ and

$1+L(w_{1})\tau+L(w_{2})\tau^{2}=\sigma(1+L(x_{1})\tau+L(x_{2})\tau^{2})\sigma^{-1}L(v)$
$=L(v)+\sigma L(x_{1})\tau\sigma^{-1}L(v)+\sigma
L(x_{2})\tau^{2}\sigma^{-1}L(v)$
$=L(v)+L(\sigma(x_{1})\tau(v))\tau+L(\sigma(x_{2})\tau^{2}(v))\tau^{2}.$

By Corollary 2.3, this is equivalent to

$L(v)=1,w_{1}=\sigma(x_{1})\tau(v),w_{2}=\sigma(x_{2})\tau^{2}(v).$

This is equivalent to $v=1$, $w_{1}=\sigma(x_{1})$, and $w_{2}=\sigma(x_{2}).$
(6) Here, $K^{(L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(L(z_{1})\tau+L(z_{2})\tau^{2},g^{\prime})}$ if and only if there are
$u,v\in K^{\times}$, $g^{\prime}=L(u^{-1})\sigma g\sigma^{-1}L(uv)$ and
$z_{1}=\tau(u)\tau(v)v^{-1}\sigma(y_{1}),z_{2}=\tau^{2}(u)\tau^{2}(v)v^{-1}\sigma(y_{2}).$
If $N(u)=1$ we get $z_{1}=y_{1}$ since $y_{1},z_{1}\in M$, and consequently
$y_{1}=\tau(u)\tau(v)v^{-1}\sigma(y_{1})$.
(7) We know that $K^{(L(y_{2})\tau^{2},g)}\cong
K^{(L(z_{2})\tau^{2},g^{\prime})}$ if and only if there are $u,v\in
K^{\times}$, $g^{\prime}=L(u^{-1})\sigma g\sigma^{-1}L(uv)$ and

$z_{2}=\tau^{2}(u)\tau^{2}(v)v^{-1}\sigma(y_{2}).$

This implies that $N(z_{2})=N(u)N(y_{2})$.

If $N(u)=1$ we get $z_{2}=y_{2}$ since $y_{2},z_{2}\in M$, and consequently
$y_{2}=\tau^{2}(u)\tau^{2}(v)v^{-1}\sigma(y_{2})$.
(8) Now $K^{(L(y_{1})\tau,g)}\cong K^{(L(z_{1})\tau,g^{\prime})}$ if and only
if there are $u,v\in K^{\times}$, $g^{\prime}=L(u^{-1})\sigma
g\sigma^{-1}L(uv)$ and

$z_{1}=\tau(u)\tau(v)v^{-1}\sigma(y_{1}).$

This implies that $N(z_{1})=N(u)N(y_{1})$.

If $N(u)=1$ we get $z_{1}=y_{1}$ since $y_{1},z_{1}\in M$ and so
$y_{1}=\tau(u)\tau(v)v^{-1}\sigma(y_{1})$. ∎

We observe that for $g=L(x_{0})+L(x_{1})\tau+L(x_{2})\tau^{2}$,
$g^{\prime}=L(w_{0})+L(w_{1})\tau+L(w_{2})\tau^{2}$, in cases (i), (ii),
(iii),

$g^{\prime}=a\sigma
g\sigma^{-1}=a(L(\sigma(x_{0}))+L(\sigma(x_{1}))\tau+L(\sigma(x_{2}))\tau^{2})$

if and only if

$w_{i}=a\sigma(x_{i})\text{ for }i\in\\{0,1,2\\}$

and that

$g^{\prime}=L(u^{-1})\sigma
g\sigma^{-1}=a(L(\sigma(x_{0}))+L(\sigma(x_{1}))\tau+L(\sigma(x_{2}))\tau^{2})L(uv)$

if and only if

$w_{i}=u^{-1}\sigma(x_{i})\tau^{i}(u)\tau^{i}(v).$

##  5\. Towards a classifications of algebras whose unital heart is a cyclic
Galois field extension

Let $A$ be a division algebra over $F$ of dimension $n$ with a unital heart
isomorphic to a cyclic Galois field extension $K$, ${\rm
Gal}(K/F)=\langle\tau\rangle$. The methology is the same as demonstrated in
the cubic field case: by Theorem 1.4, $A\cong K^{(f,g)}$ for some $(f,g)\in
G_{0}$, and by Theorem 3.3 we are left with a list of cases some of which we
can refine. Write
$f^{\prime}=L(e)+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1}$,
$e\in\\{1,0\\}$. When $e=1$ and there exists $y_{i}\in K^{\times}$ in the
expression of $f^{\prime}$ such that $\tau^{i}$ generates $G$, choose the
smallest such $y_{i}$ and scale it to an element in $M$.
(1) $f^{\prime}=id+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1}$
and all $y_{i}\in K^{\times}$ such that
$N_{A_{0}/F}(y)=N_{A_{0}/F}(1+y_{1}t+\dots+y_{n-1}t^{n-1})\not=0$: Then there
exists $y_{1}^{\prime}\in M$, such that $N(y_{1})=N(y_{1}^{\prime})$ and $v\in
K^{\times}$ with $y_{1}^{\prime}=\tau(v)v^{-1}y_{1}$. Set $\sigma=id$ and
$u=1$ in Lemma 3.1. Then put $g^{\prime}=gL(v)$ and for
$y_{i}^{\prime}=\tau^{i}(u)\tau^{i}(v)v^{-1}y_{i}$ we have

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1},g)}\cong
K^{(1+L(y^{\prime})\tau+L(y^{\prime}_{2})\tau^{2}+\dots+L(y^{\prime}_{n-1})\tau^{n-1},g^{\prime})}$

(2) $f^{\prime}=id+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1}$
for some nonempty set $\\{i_{1},\dots,i_{s}\\}\subset\\{1,\dots,n-1\\}$ of
less than $n-1$ elements, such that $y_{i_{1}},\dots,y_{i_{s}}$ equal 0,
$y_{i}\in\\{1,\dots,n-1\\}\setminus\\{i_{1},\dots,i_{s}\\}$ are nonzero, and
$N_{A_{0}/F}(1+y_{1}t+\dots+y_{n-1}t^{n-1})\not=0$:
Case 1: there exists $i_{k}$ not in $\\{i_{1},\dots,i_{s}\\}$ such that
$\tau^{i_{k}}$ generates ${\rm Gal}(K/F)$. Choose the smallest such $i_{k}$,
that means $y_{i_{k}}$ is nonzero. By Hilbert’s Theorem 90,
$S(K)=\\{\tau^{i_{k}}(v)v^{-1},|\,v\in K^{\times}\\}$. Thus there exists
$y_{i_{k}}^{\prime}\in M$ and $v\in K^{\times}$, such that
$y_{i_{k}}^{\prime}=\tau^{i_{k}}(v)v^{-1}y_{i_{k}}$. Set $\sigma=id$ and $u=1$
in Lemma 3.1. For $g^{\prime}=gL(v)$ and for

$y_{1}^{\prime}=\tau^{2}(v)v^{-1}y_{1},\dots,y_{i_{k}}^{\prime}=\tau(v)v^{-1}y_{i_{k}},\dots,y_{n-1}^{\prime}=\tau^{n-1}(v)v^{-1}y_{n-1}$

we have

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1},g)}\cong
K^{(1+L(y^{\prime})\tau+L(y^{\prime}_{2})\tau^{2}+\dots+L(y^{\prime}_{n-1})\tau^{n-1},g^{\prime})}$

This implies that

$K\cong K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1},g)}$

for $y_{i_{k}}\in M$, and for $y_{i}\in K$, such that for some nonempty set
$\\{i_{1},\dots,i_{s}\\}\subset\\{1,\dots,n-1\\}$ of less than $n-1$ elements,
the $y_{i_{1}},\dots,y_{i_{s}}$ equal 0,
$N_{A_{0}}(1+y_{1}t+\dots+y_{n-1}t^{n-1})\not=0$, and $g\in{\rm Gl(V)}$.
Case 2: for all $i_{k}$ not in $\\{i_{1},\dots,i_{s}\\}$, $\tau^{i_{k}}$ does
not generate ${\rm Gal}(K/F)$. Then we cannot “scale” any entry $y_{i}$ so
that it lies in $M$. This case can only happen when $n$ is not prime.
(3) $f^{\prime}=id$: Then $g=L(x_{0})+L(x_{1})\tau+\dots+L(x_{n-1})\tau^{n-1}$
for $x_{i}\in K$ with $N_{A_{0}}(x_{0}+x_{1}t+\dots+x_{n-1}t^{n-1})\not=0$. We
have two cases to consider: $x_{0}=0$, then choose $v=1$, and the case
$x_{0}\in K^{\times}$, then choose $v=x_{0}^{-1}$. Apply Lemma 3.1 for $u=1$,
$v$, $\sigma=id$ to see that for $g^{\prime}=gL(v)$ we obtain $K^{(1,g)}\cong
K^{(1,g^{\prime})}$. Now
$g^{\prime}=(L(e)+L(x_{1})\tau+\dots+L(x_{n-1})\tau^{n-1}$ for $e,x_{i}\in K$
with $N_{A_{0}}(e+x_{1}t+\dots+x_{n-1}t^{n-1})\not=0$ and $e\in\\{0,1\\}$.
If $e=0$ then $g^{\prime}=L(x_{1})\tau+\dots+x_{n-1}t^{n-1}$ for $x_{i}\in K$
with $N_{A_{0}}(x_{1}t+\dots+x_{n-1}t^{n-1})\not=0$.
If $e=1$ then $g^{\prime}=1+L(x_{1})\tau+\dots+x_{n-1}t^{n-1}$ for $x_{i}\in
K$ with $N_{A_{0}}(1+x_{1}t+\dots+x_{n-1}t^{n-1})\not=0$.
(4) $f^{\prime}=L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1}$ for
some $y_{i}\in K^{\times}$ such that
$N_{A_{0}/F}(y_{1}t+\dots+y_{n-1}t^{n-1})\not=0$.
(5) $f^{\prime}=L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1}$ for
some nonempty set $\\{i_{1},\dots,i_{s}\\}\subset\\{1,\dots,n-1\\}$ of less
than $n-1$ elements, such that $y_{i_{1}},\dots,y_{i_{s}}$ equal 0,
$y_{i}\in\\{1,\dots,n-1\\}\setminus\\{i_{1},\dots,i_{s}\\}$ are nonzero, and
$N_{A_{0}/F}(y_{1}t+\dots+y_{n-1}t^{n-1})\not=0$: These cases we cannot refine
any more.

By Proposition 3.1, two algebras $A$ and $B$ that belong to different types
(1), (2) and different subsets $\\{i_{1},\dots,i_{s}\\}$, and (5) with
different subsets $\\{i_{1},\dots,i_{s}\\}$, cannot be isomorphic. The same
holds for algebras that lie in either (3) or (4), again these are not
isomorphic to any of the types listed in (1), (2, $\\{i_{1},\dots,i_{s}\\}$)
and (5, $\\{i_{1},\dots,i_{s}\\})$. Since
$(A^{(f,g)})^{op}\cong(A^{op})^{(g,f)}$ we can apply Proposition 3.1 to the
opposite algebras as well which shows us that algebras of type (3) and (4)
also are not isomorphic. We proved:

######  Theorem 5.1.

(Enumeration Theorem for unital hearts that are a cyclic field extension) Let
$A$ be a division algebra of dimension $n$ with unital heart isomorphic to a
cyclic Galois field extension $K/F$ with ${\rm Gal}(K/F)=\langle\tau\rangle$.
Then $A$ is isomorphic to precisely one of the following:

  1. (1)

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1},g)}$ for
$y_{1}\in M$, $y_{i}\in K^{\times}$ such that
$N_{A_{0}}(1+y_{1}t+\dots+y_{n-1}t^{n-1})\not=0$ and some $g\in{\rm Gl(V)}$.

  2. (2)

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1},g)}$ and there
exists a nonempty set $\\{i_{1},\dots,i_{s}\\}\subset\\{1,\dots,n-1\\}$ of
less than $n-1$ elements, such that $y_{i_{1}},\dots,y_{i_{s}}$ equal 0,
$y_{i}\in\\{1,\dots,n-1\\}\setminus\\{i_{1},\dots,i_{s}\\}$ are nonzero, and
$N_{A_{0}/F}(1+y_{1}t+\dots+y_{n-1}t^{n-1})\not=0$, $g\in{\rm Gl(V)}$.
If all the $\tau^{i_{k}}$ that generate ${\rm Aut}(K)=\langle\tau\rangle$ lie
in $\\{i_{1},\dots,i_{s}\\}$ then we cannot “scale” any nonzero entry $y_{i}$
so that it lies in $M$.
If there exists $y_{i_{k}}\not=0$, such that $\tau^{i_{k}}$ generates ${\rm
Aut}_{F}(K)=\langle\tau\rangle$, then take the smallest such $y_{i_{k}}\in M$,
and get $K\cong
K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1},g)}$ for
$y_{i_{k}}\in M$.
Each such set $\\{i_{1},\dots,i_{s}\\}$ is a different type.

  3. (3)

$K^{(1,g)}$ for $g=L(x_{1})\tau+\dots+L(x_{n-1})\tau^{n-1}$ for $x_{i}\in K$
with $N_{A_{0}}(x_{1}t+\dots+x_{n-1}t^{n-1})\not=0$.

  4. (4)

$K^{(1,g)}$ for $g=1+L(x_{1})\tau+\dots+L(x_{n-1})\tau^{n-1}$ for $x_{i}\in K$
with $N_{A_{0}}(1+x_{1}t+\dots+x_{n-1}t^{n-1})\not=0$.

  5. (5)

$K^{(L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1},g)}$ and there
exists a nonempty set $\\{i_{1},\dots,i_{s}\\}\subset\\{1,\dots,n-1\\}$ of
less than $n-1$ elements, such that $y_{i_{1}},\dots,y_{i_{s}}$ equal 0,
$y_{i}\in\\{1,\dots,n-1\\}\setminus\\{i_{1},\dots,i_{s}\\}$ are nonzero, and
$N_{A_{0}/F}(y_{1}t+\dots+y_{n-1}t^{n-1})\not=0$, $g\in{\rm Gl(V)}$.
Each such set $\\{i_{1},\dots,i_{s}\\}$ is a different type.

To compete the classification of $n$-dimensional algebras which have a cyclic
Galois field extension as unital heart, we now check when two algebras of the
same type are isomorphic:

######  Theorem 5.2.

(Classification Theorem for algebras with a cyclic Galois extension as unital
heart) Let $K/F$ be a cyclic Galois field extension of degree $n$, ${\rm
Gal}(K/F)=\langle\tau\rangle$, $\sigma\in{\rm Gal}(K/F)$.

  1. (1)

For $y_{1},z_{1}\in M$, $y_{i},z_{i}\in K^{\times}$ with $N_{A_{0}}(y)\not=0$,
$N_{A_{0}}(z)\not=0$, and $g,g^{\prime}\in{\rm Gl}(V)$, we have

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1},g)}\cong
K^{(1+L(z_{1})\tau+L(z_{2})\tau^{2}+\dots+L(z_{n-1})\tau^{n-1},g^{\prime})}$

if and only if $y_{1}=z_{1}$, and exactly one of the following holds:
(i) there exists $a\in F^{\times}$, such that $g^{\prime}=ag$ and
$z_{i}=y_{i}$,
(ii) there exists $a\in F^{\times}$, such that
$z_{i}=\tau^{i}(y_{1}^{-1})y_{1}\sigma(y_{i})$, $g^{\prime}=a\tau
g\tau^{-1}L(y_{1}^{-1})$,
(iii) there exists $v\in K^{\times}$ such that
$y_{1}=\tau(v)v^{-1}\sigma(y_{1})$, $z_{i}=\tau^{i}(v)v^{-1}\sigma(y_{i})$ and
$g^{\prime}=\sigma g\sigma^{-1}L(v)$ for some $\sigma\in{\rm Gal}(K/F)$,
$\sigma\not\in\\{id,\tau\\}$.

  2. (2)

For a nonempty set $\\{i_{1},\dots,i_{s}\\}\subset\\{1,\dots,n-1\\}$ of less
than $n-1$ elements, such that the $y_{i},z_{i}\in\\{i_{1},\dots,i_{s}\\}$
equal 0, $y_{i},z_{i}\in\\{1,\dots,n-1\\}\setminus\\{i_{1},\dots,i_{s}\\}$ are
nonzero, and $N_{A_{0}/F}(1+y_{1}t+\dots+y_{n-1}t^{n-1})\not=0$,
$g,g^{\prime}\in{\rm Gl(V)}$,

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1},g)}\cong
K^{1+L(z_{1})\tau+L(z_{2})\tau^{2}+\dots+L(z_{n-1})\tau^{n-1},g^{\prime})}$

if and only if $g^{\prime}=\sigma g\sigma^{-1}L(v)$ and
$z_{i}=\tau^{i}(v)v^{-1}\sigma(y_{i})$ for all $i$. In particular, if there
exists $y_{i_{k}}\not=0$, such that $\tau^{i_{k}}$ generates ${\rm Gal}(K/F)$,
then we have the smallest such $i_{k}$ with $y_{i_{k}},z_{i_{k}}\in M$, so
that

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1},g)}\cong
K^{1+L(z_{1})\tau+L(z_{2})\tau^{2}+\dots+L(z_{n-1})\tau^{n-1},g^{\prime})}$

if and only if $y_{i_{k}}=z_{i_{k}}$, and and exactly one of the following
holds:
(i) there exists $a\in F^{\times}$, such that $g^{\prime}=ag$ and
$z_{i}=y_{i}$,
(ii) there exists $a\in F^{\times}$ such that
$z_{i}=\tau^{i}(y_{i_{k}}^{-1})y_{1}\sigma(y_{i})$ for all $i\not={i_{k}}$,
$g^{\prime}=a\tau g\tau^{-1}L(y_{i_{k}}^{-1})$,
(iii) there exists $v\in K^{\times}$ such that
$y_{i_{k}}=\tau(v)v^{-1}\sigma(y_{i_{k}})$,
$z_{i}=\tau^{i}(v)v^{-1}\sigma(y_{i})$ for all $i\not={i_{k}}$ and
$g^{\prime}=\sigma g\sigma^{-1}L(v)$ for some $\sigma\in{\rm Gal}(K/F)$,
$\sigma\not\in\\{id,\tau^{i_{k}}\\}$.

  3. (3)

For $x_{i},w_{i}\in K$ with $N_{A_{0}}(x_{1}t+\dots+x_{n-1}t^{n-1})\not=0$,
$N_{A_{0}}(w_{1}t+\dots+w_{n-1}t^{n-1})\not=0$,

$K^{(1,L(x_{1})\tau+\dots+L(x_{n-1})\tau^{n-1})}\cong
K^{(1,L(w_{1})\tau+\dots+L(w_{n-1})\tau^{n-1})}$

if and only if $w_{i}=\sigma(x_{i})\tau^{i}(v)$ for all $i$.

  4. (4)

For $x_{i},w_{i}\in K$ with $N_{A_{0}}(1+x_{1}t+\dots+x_{n-1}t^{n-1})\not=0$,
$N_{A_{0}}(1+w_{1}t+\dots+w_{n-1}t^{n-1})\not=0$,

$K^{(1,1+L(x_{1})\tau+\dots+L(x_{n-1})\tau^{n-1})}\cong
K^{(1,1+L(w_{1})\tau+\dots+L(w_{n-1})\tau^{n-1})}$

if and only if $v=1$ and $w_{i}=\sigma(x_{i})\tau^{i}(v)$ for all $i$.

  5. (5)

For a nonempty set $\\{i_{1},\dots,i_{s}\\}\subset\\{1,\dots,n-1\\}$ of less
than $n-1$ elements, such that the $y_{i},z_{i}\in\\{i_{1},\dots,i_{s}\\}$
equal 0, $y_{i},z_{i}\in\\{1,\dots,n-1\\}\setminus\\{i_{1},\dots,i_{s}\\}$ are
nonzero, and $N_{A_{0}/F}(y_{1}t+\dots+y_{n-1}t^{n-1})\not=0$,
$N_{A_{0}/F}(z_{1}t+\dots+z_{n-1}t^{n-1})\not=0$, $g,g^{\prime}\in{\rm
Gl(V)}$,

$K^{(L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1},g)}\cong
K^{(L(z_{1})\tau+L(z_{2})\tau^{2}+\dots+L(z_{n-1})\tau^{n-1},g^{\prime})}$

if and only if there are are $u,v\in K^{\times}$ such that
$g^{\prime}=L(u^{-1})\sigma g\sigma^{-1}L(uv)$ and
$z_{i}=\tau^{i}(u)\tau^{i}(v)v^{-1}\sigma(y_{i})$ for all
$y_{i},z_{i}\in\\{1,\dots,n-1\\}\setminus\\{i_{1},\dots,i_{s}\\}$.

###### Proof.

By Lemma 3.1, two algebras mentioned in any of the cases of Theorem 5.2 are
isomorphic if and only if there are $u,v\in K^{\times}$ and $\sigma\in{\rm
Gal}(K/F)$, such that the critical relations in Lemma 3.1 are satisfied.
(1) The critical relations are $z_{i}=\tau^{i}(v)v^{-1}\sigma(y_{i})$ and
$g^{\prime}=\sigma g\sigma^{-1}L(v)$, which implies
$N_{K}(z_{1})=N_{K}(y_{1})$, hence $z_{1}=y_{1}$ by the definition of $M$
(Lemma 3.7). Using $z_{1}=y_{1}$ we go back to the equation
$z_{1}=\tau(v)v^{-1}\sigma(y_{1})$. If $\sigma=id$ this yields $v=a\in
F^{\times}$. If $\sigma=\tau$ then $a=vy_{1}\in F^{\times}$. Therefore the
critical relations hold if and only if $z_{1}=y_{1}$ and there exists $a\in
F^{\times}$ such that $g^{\prime}=ag$ and $z_{i}=y_{i}$, or there exists $a\in
F^{\times}$ such that $z_{i}=\tau^{i}(y_{1}^{-1})y_{1}\sigma(y_{i})$ and
$g^{\prime}=a\tau g\tau^{-1}L(y_{1}^{-1})$, or
$y_{1}=\tau(v)v^{-1}\sigma(y_{1})$, $z_{i}=\tau^{i}(v)v^{-1}\sigma(y_{i})$ and
$g^{\prime}=\sigma g\sigma^{-1}L(v)$ for some $\sigma\in{\rm Gal}(K/F)$,
$\sigma\not\in\\{id,\tau\\}$.
(2) The critical relations are $g^{\prime}=\sigma g\sigma^{-1}L(v)$ and
$z_{i}=\tau^{i}(v)v^{-1}\sigma(y_{i})$. If there exists $y_{i_{k}}\not=0$,
such that $\tau^{i_{k}}$ generates ${\rm Gal}(K/F)$, then $K\cong
K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1},g)}$ for
$y_{i_{k}}\in M$. Then $N_{K}(z_{i_{k}})=N_{K}(y_{i_{k}})$, hence
$z_{i_{k}}=y_{i_{k}}$ by the definition of $M$. This yields $v=a\in
F^{\times}$ when $\sigma=id$ and $v=ay_{i_{k}}^{-1}$ when
$\sigma=\tau^{i_{k}}$.
(3) Here, the critical relations reduce to $u=1$ and

$L(w_{1})\tau+\dots+L(w_{n-1})\tau^{n-1}=\sigma(L(x_{1})\tau+\dots+L(x_{n-1})\tau^{n-1})\sigma^{-1}L(v)$
$=\sigma L(x_{1})\tau\sigma^{-1}L(v)+\sigma
L(x_{2})\tau^{2}\sigma^{-1}L(v)+\dots+\sigma
L(x_{2})\tau^{n-1}\sigma^{-1}L(v)$
$=L(\sigma(x_{1})\tau(v))\tau+L(\sigma(x_{2})\tau^{2}(v))\tau^{2}+\dots+L(\sigma(x_{2})\tau^{n-1}(v))\tau^{n-1}.$

By Corollary 2.3, this is equivalent to $w_{i}=\sigma(x_{i})\tau^{i}(v)$ for
all $i$.
(4) Here, the critical relations reduce to $u=1$ and

$1+L(w_{1})\tau+\dots+L(w_{n-1})\tau^{n-1}=\sigma(1+L(x_{1})\tau+\dots+L(x_{n-1})\tau^{n-1})\sigma^{-1}L(v)$
$=L(v)+\sigma L(x_{1})\tau\sigma^{-1}L(v)+\dots+\sigma
L(x_{n-1})\tau^{n-1}\sigma^{-1}L(v)$
$=L(v)+L(\sigma(x_{1})\tau(v))\tau+\dots+L(\sigma(x_{n-1})\tau^{n-1}(v))\tau^{n-1}.$

By Corollary 2.3, this is equivalent to $L(v)=1$ and
$w_{i}=\sigma(x_{i})\tau^{i}(v)$ for all $i$. This is equivalent to $v=1$ and
$w_{i}=\sigma(x_{i})\tau^{i}(v)$ for all $i$.
(5) Here, the critical relations cannot be simplified. ∎

######  Corollary 5.3.

(Classification Theorem for algebras with a cyclic Galois extension of prime
degree as unital heart) Let $K/F$ be a cyclic Galois field extension of prime
degree $n$ with ${\rm Gal}(K/F)=\langle\tau\rangle$, $\sigma\in{\rm
Gal}(K/F)$.

  1. (1)

For $y_{1},z_{1}\in M$, $y_{i},z_{i}\in K^{\times}$ with $N_{A_{0}}(y)\not=0$,
$N_{A_{0}}(z)\not=0$, and $g,g^{\prime}\in{\rm Gl}(V)$, we have

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1},g)}\cong
K^{(1+L(z_{1})\tau+L(z_{2})\tau^{2}+\dots+L(z_{n-1})\tau^{n-1},g^{\prime})}$

if and only if $y_{1}=z_{1}$, and exactly one of the following holds:
(i) there exists $a\in F^{\times}$, such that $g^{\prime}=ag$ and
$z_{i}=y_{i}$,
(ii) there exists $a\in F^{\times}$ such that
$z_{i}=\tau^{i}(y_{1}^{-1})y_{1}\sigma(y_{i})$, $g^{\prime}=a\tau
g\tau^{-1}L(y_{1}^{-1})$,
(iii) there exists $v\in K^{\times}$ such that
$y_{1}=\tau(v)v^{-1}\sigma(y_{1})$, $z_{i}=\tau^{i}(v)v^{-1}\sigma(y_{i})$ and
$g^{\prime}=\sigma g\sigma^{-1}L(v)$ for some $\sigma\in{\rm Gal}(K/F)$,
$\sigma\not\in\\{id,\tau\\}$.

  2. (2)

For a nonempty set $\\{i_{1},\dots,i_{s}\\}\subset\\{1,\dots,n-1\\}$ of less
than $n-1$ elements, such that the $y_{i},z_{i}\in\\{i_{1},\dots,i_{s}\\}$
equal 0, $y_{i},z_{i}\in\\{1,\dots,n-1\\}\setminus\\{i_{1},\dots,i_{s}\\}$ are
nonzero, and $N_{A_{0}/F}(1+y_{1}t+\dots+y_{n-1}t^{n-1})\not=0$,
$g,g^{\prime}\in{\rm Gl(V)}$, and let $i_{k}$ be such that
$y_{i_{k}},z_{i_{k}}\in M$ (i.e. this is the smallest $i_{k}$ be such that
$y_{i_{k}}\not=0$). Then

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1},g)}\cong
K^{1+L(z_{1})\tau+L(z_{2})\tau^{2}+\dots+L(z_{n-1})\tau^{n-1},g^{\prime})}$

if and only if $y_{i_{k}}=z_{i_{k}}$, and exactly one of the following holds:
(i) there exists $a\in F^{\times}$, such that $g^{\prime}=ag$ and
$z_{i}=y_{i}$,
(ii) there exists $a\in F^{\times}$ such that
$z_{i}=\tau^{i}(y_{i_{k}}^{-1})y_{1}\sigma(y_{i})$ for all $i\not={i_{k}}$,
$g^{\prime}=a\tau g\tau^{-1}L(y_{i_{k}}^{-1})$,
(iii) there exists $v\in K^{\times}$ such that
$y_{i_{k}}=\tau(v)v^{-1}\sigma(y_{i_{k}})$,
$z_{i}=\tau^{i}(v)v^{-1}\sigma(y_{i})$ for all $i\not={i_{k}}$ and
$g^{\prime}=\sigma g\sigma^{-1}L(v)$ for some $\sigma\in{\rm Gal}(K/F)$,
$\sigma\not\in\\{id,\tau^{i_{k}}\\}$.

  3. (3)

For $x_{i},w_{i}\in K$ with $N_{A_{0}}(x_{1}t+\dots+x_{n-1}t^{n-1})\not=0$,
$N_{A_{0}}(w_{1}t+\dots+w_{n-1}t^{n-1})\not=0$,

$K^{(1,L(x_{1})\tau+\dots+L(x_{n-1})\tau^{n-1})}\cong
K^{(1,L(w_{1})\tau+\dots+L(w_{n-1})\tau^{n-1})}$

if and only if $w_{i}=\sigma(x_{i})\tau^{i}(v)$ for all $i$.

  4. (4)

For $x_{i},w_{i}\in K$ with $N_{A_{0}}(1+x_{1}t+\dots+x_{n-1}t^{n-1})\not=0$,
$N_{A_{0}}(1+w_{1}t+\dots+w_{n-1}t^{n-1})\not=0$,

$K^{(1,1+L(x_{1})\tau+\dots+L(x_{n-1})\tau^{n-1})}\cong
K^{(1,1+L(w_{1})\tau+\dots+L(w_{n-1})\tau^{n-1})}$

if and only if $v=1$ and $w_{i}=\sigma(x_{i})\tau^{i}(v)$ for all $i$.

  5. (5)

For a nonempty set $\\{i_{1},\dots,i_{s}\\}\subset\\{1,\dots,n-1\\}$ of less
than $n-1$ elements, such that the $y_{i},z_{i}\in\\{i_{1},\dots,i_{s}\\}$
equal 0, $y_{i},z_{i}\in\\{1,\dots,n-1\\}\setminus\\{i_{1},\dots,i_{s}\\}$ are
nonzero, and $N_{A_{0}/F}(y_{1}t+\dots+y_{n-1}t^{n-1})\not=0$,
$N_{A_{0}/F}(z_{1}t+\dots+z_{n-1}t^{n-1})\not=0$, $g,g^{\prime}\in{\rm
Gl(V)}$,

$K^{(L(y_{1})\tau+L(y_{2})\tau^{2}+\dots+L(y_{n-1})\tau^{n-1},g)}\cong
K^{(L(z_{1})\tau+L(z_{2})\tau^{2}+\dots+L(z_{n-1})\tau^{n-1},g^{\prime})}$

if and only if there are are $u,v\in K^{\times}$ such that
$g^{\prime}=L(u^{-1})\sigma g\sigma^{-1}L(uv)$ and
$z_{i}=\tau^{i}(u)\tau^{i}(v)v^{-1}\sigma(y_{i})$ for all
$y_{i},z_{i}\in\\{1,\dots,n-1\\}\setminus\\{i_{1},\dots,i_{s}\\}$.

We were not able to refine the conditions in (5) any further.

A list of nonisomorphic algebras which have a $C_{n}$-algebra as unital heart
can be done as well along the same lines as we will demonstrate in the next
section for the three-dimensional case. However, it will be much less “tight”
for larger $n$.

##  6\. A classification of algebras whose unital heart is a $C_{3}$-Galois
algebra

Let $K\in{\rm Alg}(V)$ be a $C_{3}$-Galois algebra with automorphism group
generated by $\tau$ and write $N_{A_{0}}(y)=N_{A_{0}/F}(1+y_{1}t+y_{2}t^{2})$.
Let $(f,g)\in G_{0}$. Let $A$ be a regular algebra over $F$ of dimension $n$
with a unital heart isomorphic to $K$. By Theorem 1.4, $A\cong K^{(f,g)}$ for
some $(f,g)\in G_{0}$.

######  Theorem 6.1.

(Enumeration Theorem for algebras with a $C_{3}$-Galois algebra as a unital
heart) Let $A$ be an algebra of dimension three with unital heart isomorphic
to a $C_{3}$-Galois algebra. Let $g\in{\rm Gl(V)}$. Then $A$ is isomorphic to
exactly one of the following types:

  1. (1)

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2},g)}$ for $y_{1}\in M$, $y_{2}\in
K^{\times}$ with $N_{A_{0}}(y)\not=0$.

  2. (2)

$K^{(1+L(y_{2})\tau^{2},g)}$ for $y_{2}\in M$, such that
$N_{K}(y_{2})\not=-1$.

  3. (3)

$K^{(1+L(y_{1})\tau,g)}$ for $y_{1}\in M$, such that $N_{K}(y_{1})\not=-1$.

  4. (4)

$K^{(L(y_{1})\tau+L(y_{2})\tau^{2},g)}$ for $y_{1}\in M$, $y_{2}\in
K^{\times}$, such that $N_{K}(y_{2})\not=-N_{K}(y_{1})$.

  5. (5)

$K^{(L(y_{2})\tau^{2},g)}$ for $y_{2}\in M$ with $N_{K}(y_{2})\not=0$.

  6. (6)

$K^{(L(y_{1})\tau,g)}$ for $y_{1}\in M$ with $N_{K}(y_{1})\not=0$.

  7. (7)

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2},g)}$ for some $0\not=y_{i}\in K\setminus
K^{\times}$ with $N_{A_{0}}(1+y_{1}t+y_{2}t^{2})\not=0$.

  8. (8)

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2},g)}$ for some $0\not=y_{1}\in K\setminus
K^{\times}$, $y_{2}\in M$ with $N_{A_{0}}(1+y_{1}t+y_{2}t^{2})\not=0$:

  9. (9)

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2},g)}$ for $y_{1}\in M$, $y_{2}\in
K\setminus K^{\times}$ such that $N_{A_{0}}(1+y_{1}t+y_{2}t^{2})\not=0$.

  10. (10)

$K^{(L(y_{1})\tau+L(y_{2})\tau^{2},g)}$ for some $0\not=y_{i}\in K\setminus
K^{\times}$ with $N_{A_{0}}(y_{1}t+y_{2}t^{2})\not=0$.

  11. (11)

$K^{(L(y_{1})\tau+L(y_{2})\tau^{2},g)}$ for $y_{2}\in M$, $y_{1}\in K\setminus
K^{\times}$ such that $N_{A_{0}}(y_{1}t+y_{2}t^{2})\not=0$.

  12. (12)

$K^{(L(y_{1})\tau+L(y_{2})\tau^{2},g)}$ for $y_{1}\in M$, $y_{2}\in K\setminus
K^{\times}$ with $N_{A_{0}}(y_{1}t+y_{2}t^{2})\not=0$.

  13. (13)

$K^{(L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2},g)}$for $y_{1}\in M$, $y_{2}\in
K^{\times}$ with $N_{A_{0}}(y_{0}+y_{1}t+y_{2}t^{2})\not=0$.

  14. (14)

$K^{(L(y_{0})+L(y_{2})\tau^{2},g)}$ for some $0\not=y_{0}\in K\setminus
K^{\times}$ unequal to 1, $y_{2}\in M$ with
$N_{A_{0}}(y_{0}+y_{2}t^{2})\not=0$.

  15. (15)

$K^{(L(y_{0})+L(y_{1})\tau,g)}$ for some $0\not=y_{0}\in K\setminus
K^{\times}$ unequal to 1, $y_{1}\in M$ with $N_{A_{0}}(y_{0}+y_{1}t)\not=0$.

  16. (16)

$K^{(L(y_{0})+L(y_{2})\tau^{2},g)}$ for some $0\not=y_{0}\in K\setminus
K^{\times}$ unequal to 1, $0\not=y_{2}\in K\setminus K^{\times}$ with
$N_{A_{0}}(y_{0}+y_{2})\not=0$.

  17. (17)

$K^{(L(y_{0})+L(y_{1})\tau,g)}$. for some $0\not=y_{0}\in K\setminus
K^{\times}$ unequal to 1, $0\not=y_{1}\in K\setminus K^{\times}$ with
$N_{A_{0}}(y_{0}+y_{1})\not=0$.

  18. (18)

$K^{(L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2},g)}$ for some $0\not=y_{0}\in
K\setminus K^{\times}$ unequal to 1, $0\not=y_{i}\in K\setminus K^{\times}$
with $N_{A_{0}}(y_{0}+y_{1}t+y_{2}t^{2})\not=0$.

  19. (19)

$K^{(L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2},g)}$ for some $0,1\not=y_{0}\in
K\setminus K^{\times}$, $0\not=y_{1}\in K\setminus K^{\times}$, $y_{2}\in M$
with $N_{A_{0}}(e+y_{1}t+y_{2}t^{2})\not=0$.

  20. (20)

$K^{(L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2},g)}$ for some $0,1\not=y_{0}\in
K\setminus K^{\times}$, $y_{1}\in M$, $0\not=y_{2}\in K\setminus K^{\times}$
with $N_{A_{0}}(e+y_{1}t+y_{2}t^{2})\not=0$.

  21. (21)

$K^{(1,g)}$ with $g=L(x_{1})\tau+L(x_{2})\tau^{2}$ for $x_{i}\in K$ with
$N_{A_{0}}(x_{1}t+x_{2}t^{2})\not=0$.

  22. (22)

$K^{(1,g)}$ with $g=1+L(x_{1})\tau+L(x_{2})\tau^{2}$ for $x_{i}\in K$ with
$N_{A_{0}}(1+x_{1}t+x_{2}t^{2})\not=0$.

  23. (23)

$K^{(1,g)}$ with $g=L(x_{0})+L(x_{1})\tau+L(x_{2})\tau^{2}$ for
$0\not=x_{0}\in K\setminus K^{\times}$ unequal to 1, $x_{i}\in K$ with
$N_{A_{0}}(x_{0}+x_{1}t+x_{2}t^{2})\not=0$.

###### Proof.

By Corollary 3.6 we have a list of types of Albert isotopes to consider, where
types (1) to (8) are already considered in Theorem 5.1 and are dealt with
analogously here. We are left with the following types:
(9) $f^{\prime}=id+L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $0\not=y_{i}\in
K\setminus K^{\times}$ with $N_{A_{0}/F}(1+y_{1}t+y_{2}t^{2})\not=0$. It seems
not possible to refine this case.
(10) $f^{\prime}=id+L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $0\not=y_{1}\in
K\setminus K^{\times}$, $y_{2}\in K^{\times}$ with
$N_{A_{0}}(1+y_{1}t+y_{2}t^{2})\not=0$: Then there exists an element
$y_{2}^{\prime}\in M$, such that $y_{2}^{\prime}=y_{2}\,{\rm mod}\,S(K)$, i.e.
there exists $v\in K^{\times}$ such that
$y_{2}^{\prime}=\tau^{2}(v)v^{-1}y_{2}$. Set $\sigma=id$, $u=1$ and
$g^{\prime}=gL(v)$ in Lemma 3.1. Then for $y_{1}^{\prime}=\tau(v)v^{-1}y_{2}$,
$y_{2}^{\prime}=\tau^{2}(v)v^{-1}y_{2}$, we have

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(1+L(y_{1}^{\prime})\tau+L(y^{\prime}_{2})\tau^{2},g^{\prime})}.$

This implies that $K\cong K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2},g)}$ for
$y_{2}\in M$, $y_{1}\in K\setminus K^{\times}$ such that
$N_{A_{0}}(y_{1}t+y_{2}t^{2})\not=0$ and some $g\in{\rm Gl(V)}$.
(11) $f^{\prime}=id+L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $y_{1}\in
K^{\times}$ $0\not=y_{2}\in K\setminus K^{\times}$ with
$N_{A_{0}}(y_{1}t+y_{2}t^{2})\not=0$: Then there exists an element
$y_{1}^{\prime}\in M$, such that $y_{1}^{\prime}=y_{1}\,{\rm mod}\,S(K)$, i.e.
there exists $v\in K^{\times}$ such that $y_{1}^{\prime}=\tau(v)v^{-1}y_{1}$,
so that $N_{K}(y_{1})=N_{K}(y_{1}^{\prime})$. Set $\sigma=id$, $u=1$ and
$g^{\prime}=gL(v)$ in Lemma 3.1. Then for
$y_{2}^{\prime}=\tau^{2}(v)v^{-1}y_{2}$ we have

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(1+L(y_{1}^{\prime})\tau+L(y^{\prime}_{2})\tau^{2},g^{\prime})}.$

This implies that $K\cong K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2},g)}$ for
$y_{1}\in M$, $y_{2}\in K\setminus K^{\times}$ such that
$N_{A_{0}}(y_{1}t+y_{2}t^{2})\not=0$ and some $g\in{\rm Gl(V)}$.
(12) $f^{\prime}=L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $0\not=y_{i}\in
K\setminus K^{\times}$ with $N_{A_{0}}(y_{1}t+y_{2}t^{2})\not=0$.
(13) $f^{\prime}=L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $0\not=y_{1}\in
K\setminus K^{\times}$, $y_{2}\in K^{\times}$ with
$N_{A_{0}}(y_{1}t+y_{2}t^{2})\not=0$: Again there exists $y_{2}^{\prime}\in
M$, such that $N_{K}(y_{2})=N_{K}(y_{2}^{\prime})$ and $v\in K^{\times}$ such
that $y_{2}^{\prime}=\tau^{2}(v)v^{-1}y_{2}$. Set $\sigma=id$ and $u=1$ in
Lemma 3.1. Then for $g^{\prime}=gL(v)$,

$K^{(L(y_{2})\tau^{2},g)}\cong K^{(L(y^{\prime}_{2})\tau^{2},g^{\prime})}.$

This implies $K\cong K^{(L(y_{1})\tau+L(y_{2})\tau^{2},g)}$ for $y_{2}\in M$,
$y_{2}\in K\setminus K^{\times}$ such that
$N_{A_{0}}(y_{1}t+y_{2}t^{2})\not=0$ and $g\in{\rm Gl(V)}$.
(14) $f^{\prime}=L(y_{1})\tau+L(y_{2})\tau^{2}$ for some $y_{1}\in K^{\times}$
$0\not=y_{2}\in K\setminus K^{\times}$ with
$N_{A_{0}}(y_{1}t+y_{2}t^{2})\not=0$: There exists $y_{1}^{\prime}\in M$, such
that $N_{K}(y_{1})=N_{K}(y_{1}^{\prime})$ and $v\in K^{\times}$ such that
$y_{1}^{\prime}=\tau(v)v^{-1}y_{1}$. Set $\sigma=id$ and $u=1$ in Lemma 3.1.
Then for $g^{\prime}=gL(v)$ and for
$y_{2}^{\prime}=\tau^{2}(v_{1})v_{1}^{-1}y_{2}$ we get

$K^{(L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(L(y_{1}^{\prime})\tau+L(y^{\prime}_{2})\tau^{2},g^{\prime})}$

by Lemma 3.1. This implies that $K\cong K^{(L(y_{1})\tau+L(y_{2})\tau^{2},g)}$
for $y_{1}\in M$, $y_{2}\in K\setminus K^{\times}$, such that
$N_{A_{0}}(y_{1}t+y_{2}t^{2})\not=0$ and $g\in{\rm Gl(V)}$. The types (15) to
(23) are treated analogously.

Analogously as in Theorem 5.1, by Lemma 3.1, two algebras $A$ and $B$ that
belong to different types (1) to (20) cannot be isomorphic. The same holds for
algebras that lie in either (21), (22) or (23), again these are not isomorphic
to any of the types listed in (1) to (20). Since
$(A^{(f,g)})^{op}\cong(A^{op})^{(g,f)}$ we can apply Lemma 3.1 to the opposite
algebras as well which shows us that algebras of type (21), (22) and (23) also
are mutually nonisomorphic. ∎

To complete the classification of three dimensional algebras which have a
$C_{3}$-Galois algebra as unital heart, we now check when two algebras of the
same type are isomorphic:

######  Theorem 6.2.

(Classification Theorem for algebras with a $C_{3}$-Galois algebra as unital
heart) Let $K/F$ be a $C_{3}$-Galois algebra with automorphism group generated
by $\tau$ and $\sigma\in\\{id,\tau,\tau^{2}\\}$. $g,g^{\prime}\in{\rm Gl(V)}$.
Additionally to the cases (1) to (8) from Theorem 4.2 we now also have:

  1. (9)

For nonzero $y_{i},z_{i}\in K\setminus K^{\times}$ with
$N_{K}(y_{1})+N_{K}(y_{2})-T_{K}(\tau(y_{1})\tau^{2}(y_{2}))\not=-1$,
$N_{K}(z_{1})+N_{K}(z_{2})-T_{K}(\tau(z_{1})\tau^{2}(z_{2}))\not=-1$, we have

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(1+L(z_{1})\tau+L(z_{2})\tau^{2},g^{\prime})}$

if and only if there is $v\in K^{\times}$, such that
$z_{1}=\tau(v)v^{-1}\sigma(y_{1})$, $z_{2}=\tau^{2}(v)v^{-1}\sigma(y_{2})$ and
$g^{\prime}=\sigma g\sigma^{-1}L(v)$ for some
$\sigma\in\\{id,\tau,\tau^{2}\\}$.

  2. (10)

For nonzero $y_{1},z_{1}\in K\setminus K^{\times}$ and $y_{2},z_{2}\in M$ such
that $N_{A_{0}}(1+y_{1}t+y_{2}t^{2})\not=0$,
$N_{A_{0}}(1+z_{1}t+z_{2}t^{2})\not=0$, we have

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(1+L(z_{1})\tau+L(z_{2})\tau^{2},g^{\prime})}$

if and only if $z_{2}=y_{2}$ and exactly one of the following holds:
(i) $z_{1}=y_{1}$ and there exists $a\in F^{\times}$ such that
$g^{\prime}=ag,$
(ii) there is $v\in K^{\times}$ such that
$z_{1}=\tau(v)v^{-1}\tau(y_{1}),y_{2}=\tau^{2}(v)v^{-1}\tau(y_{2})$ and
$g^{\prime}=\tau g\tau^{-1}L(v),$
(iii) $z_{1}=\tau(y_{2}^{-1})y_{2}\tau^{2}(y_{1})$ and there exists $a\in
F^{\times}$ such that $g^{\prime}=a\tau^{2}g\tau^{-2}L(y_{2})^{-1}.$

  3. (11)

For $y_{1},z_{1}\in M$ and nonzero $y_{2},z_{2}\in K\setminus K^{\times}$ such
that $N_{A_{0}}(1+y_{1}t+y_{2}t^{2})\not=0$,
$N_{A_{0}}(1+z_{1}t+z_{2}t^{2})\not=0$, we have

$K^{(1+L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(1+L(z_{1})\tau+L(z_{2})\tau^{2},g^{\prime})}$

if and only if $z_{1}=y_{1}$ and exactly one of the following holds:
(i) $z_{2}=y_{2}$ and there exists $a\in F^{\times}$ such that
$g^{\prime}=ag,$
(ii) $z_{2}=\tau^{2}(y_{1}^{-1})y_{1}\tau(y_{2})$ and $g^{\prime}=a\tau
g\tau^{-1}L(y_{1})^{-1},$
(iii) $z_{2}=\tau^{2}(v)v^{-1}\tau^{2}(y_{2})$ and there exists $a\in
F^{\times}$ such that $g^{\prime}=a\tau^{2}g\tau^{-2}L(y_{1})^{-1}.$

  4. (12)

For nonzero $y_{i},z_{i}\in K\setminus K^{\times}$ with
$N_{A_{0}}(y_{1}t+y_{2}t^{2})\not=0$, we have

$K^{(L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(L(z_{1})\tau+L(z_{2})\tau^{2},g^{\prime})}$

if and only if there are $u,v\in K^{\times}$, such that
$z_{1}=\tau(u)\tau(v)v^{-1}\sigma(y_{1}),$
$z_{2}=\tau^{2}(u)\tau^{2}(v)v^{-1}\sigma(y_{2})$ and
$g^{\prime}=L(u^{-1})\sigma g\sigma^{-1}L(uv)$

  5. (13)

For nonzero $y_{1},z_{1}\in K\setminus K^{\times}$ and $y_{2},z_{2}\in M$ such
that $N_{A_{0}}(y_{1}t+y_{2}t^{2})\not=0$, we have

$K^{(L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(L(z_{1})\tau+L(z_{2})\tau^{2},g^{\prime})}$

if and only if there are $u,v\in K^{\times}$, $g^{\prime}=L(u^{-1})\sigma
g\sigma^{-1}L(uv)$ and

$z_{1}=\tau(u)\tau(v)v^{-1}\sigma(y_{1}),\quad
z_{2}=\tau^{2}(u)\tau^{2}(v)v^{-1}\sigma(y_{2}).$

This implies that $N(z_{1})=N(u)N(y_{1})$ and $N(z_{2})=N(u)N(y_{2})$. If
$N(u)=1$ then $z_{2}=y_{2}$.

  6. (14)

For $y_{1},z_{1}\in M$ and nonzero $y_{2},z_{2}\in K\setminus K^{\times}$ with
$N_{A_{0}}(y_{1}t+y_{2}t^{2})\not=0$, $N_{A_{0}}(z_{1}t+z_{2}t^{2})\not=0$, we
have

$K^{(L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(L(z_{1})\tau+L(z_{2})\tau^{2},g^{\prime})}$

if and only if there are $u,v\in K^{\times}$, $g^{\prime}=L(u^{-1})\sigma
g\sigma^{-1}L(uv)$ and

$z_{1}=\tau(u)\tau(v)v^{-1}\sigma(y_{1}),\quad
z_{2}=\tau^{2}(u)\tau^{2}(v)v^{-1}\sigma(y_{2}).$

This implies that $N(z_{1})=N(u)N(y_{1})$ and $N(z_{2})=N(u)N(y_{2})$.

  7. (15)

For nonzero $y_{i},z_{i}\in K\setminus K^{\times}$, $y_{0},z_{0}\not=1$, with
$N_{A_{0}}(y_{0}+y_{1}t+y_{2}t^{2})\not=0$,
$N_{A_{0}}(z_{0}+z_{1}t+z_{2}t^{2})\not=0$ and $g\in{\rm Gl(V)}$, we have

$K^{(L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(L(z_{0})+L(z_{1})\tau+L(z_{2})\tau^{2},g)}$

if and only if there are $u,v\in K^{\times}$, such that
$z_{0}=u\sigma(y_{0}),$ $z_{i}=\tau^{i}(u)\tau^{i}(v)v^{-1}\sigma(y_{i})$ and
$g^{\prime}=L(u^{-1})\sigma g\sigma^{-1}L(uv)$.

  8. (16)

For nonzero $y_{1},z_{1}\in K\setminus K^{\times}$, $y_{0},z_{0}\not=1$,
$y_{2},z_{2}\in M$ with $N_{A_{0}}(e+y_{1}t+y_{2}t^{2})\not=0$ and $g\in{\rm
Gl(V)}$, we have

$K^{(L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(L(z_{0})+L(z_{1})\tau+L(z_{2})\tau^{2},g)}$

if and only if there are $u,v\in K^{\times}$, such that
$z_{0}=u\sigma(y_{0}),$ $z_{i}=\tau^{i}(u)\tau^{i}(v)v^{-1}\sigma(y_{i})$ and
$g^{\prime}=L(u^{-1})\sigma g\sigma^{-1}L(uv)$.

  9. (17)

For nonzero $y_{0},z_{0}\in K\setminus K^{\times}$, $y_{0},z_{0}\not=1$,
$y_{1},z_{1}\in M$, nonzero $y_{2},z_{2}\in K\setminus K^{\times}$ with
$N_{A_{0}}(y_{0}+y_{1}t+y_{2}t^{2})\not=0$,
$N_{A_{0}}(z_{0}+z_{1}t+z_{2}t^{2})\not=0$, we have

$K^{(L(y_{0})+L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(L(z_{0})+L(z_{1})\tau+L(z_{2})\tau^{2},g)}$

if and only if $g^{\prime}=\sigma g\sigma^{-1}L(v)$ and
$z_{i}=\tau^{i}(v)v^{-1}\sigma(y_{i})$ for all $i$.

  10. (18)

For $y_{0},z_{0}\in K\setminus K^{\times}$, $x_{i},w_{i}\in K$ with
$N_{A_{0}}(x_{0}+x_{1}t+x_{2}t^{2})\not=0$,
$N_{A_{0}}(w_{0}+w_{1}t+w_{2}t^{2})\not=0$, we have

$K^{(1,L(x_{0})+L(x_{1})\tau+L(x_{2})\tau^{2})}\cong
K^{(1,L(z_{0})+L(z_{1})\tau+L(z_{2})\tau^{2})}$

if and only if there are $u,v\in K^{\times}$, such that
$w_{0}=\sigma(z_{0})v,w_{1}=\sigma(x_{1})\tau(v)\tau(u)u^{-1},w_{2}=\sigma(x_{1})\tau^{2}(v)\tau^{2}(u)u^{-1}$.

###### Proof.

Cases (1) to (8) are proved analogously as in the Classification Theorem 4.2
for algebras with a cubic field extension as a unital heart. By Lemma 3.1, two
algebras mentioned in any of the cases of Theorem 3.5 are isomorphic if and
only if there are $u,v\in K^{\times}$ and $\sigma\in{\rm Aut}_{F}(K)$, such
that the critical relations in of Lemma 3.1 are satisfied.
(9) Here $u=1$ so the critical relations are
$z_{1}=\tau(v)v^{-1}\sigma(y_{1})$, $z_{2}=\tau^{2}(v)v^{-1}\sigma(y_{2})$ and
$g^{\prime}=\sigma g\sigma^{-1}L(v)$.
(10) Here $u=1$ so the critical relations $z_{1}=\tau(v)v^{-1}\sigma(y_{1})$,
$z_{2}=\tau^{2}(v)v^{-1}\sigma(y_{2})$ and $g^{\prime}=\sigma
g\sigma^{-1}L(v)$, which imply $z_{2}=y_{2}$ by the definition of $M$.

Assume $\sigma=id$. Using $z_{2}=y_{2}$, the equation
$z_{2}=\tau^{2}(v)v^{-1}\sigma(y_{2})$ yields $\tau^{2}(v)v^{-1}=1$, so
$v=a\in F^{\times}$. Therefore the critical relations are $z_{1}=y_{1}$,
$z_{2}=y_{2}$, and there exists $a\in F^{\times}$ such that $g^{\prime}=ag$.

Assume $\sigma=\tau$. Then the critical relations are
$z_{1}=\tau(v)v^{-1}\tau(y_{1})$, $z_{2}=y_{2}$,
$y_{2}=\tau^{2}(v)v^{-1}\tau(y_{2})$ and $g^{\prime}=\tau g\tau^{-1}L(v)$.

Assume $\sigma=\tau^{2}$. Then the critical relations are
$y_{2}=\tau^{2}(v)v^{-1}\tau^{2}(y_{2})$ and
$g^{\prime}=\tau^{2}g\tau^{-2}L(v)$. The first equation is the same as
$\tau^{2}(vy_{2})=vy_{2}$, i.e. $a=vy_{2}\in F^{\times}$ and
$g^{\prime}=\tau^{2}g\tau^{-2}L(v)$. Hence there is $a\in F^{\times}$, such
that $v=ay_{2}^{-1}$ and we get $z_{1}=\tau(y_{2}^{-1})y_{2}\tau^{2}(y_{1})$,
$z_{2}=y_{2}$, $g^{\prime}=a\tau^{2}g\tau^{-2}L(y_{2})^{-1}$.
(11) The critical relations are $z_{1}=\tau(v)v^{-1}\sigma(y_{1})$,
$z_{2}=\tau^{2}(v)v^{-1}\sigma(y_{2})$ and $g^{\prime}=\sigma
g\sigma^{-1}L(v)$, which imply $z_{1}=y_{1}$ by the definition of $M$.

Assume $\sigma=id$. This yields $\tau(v)v^{-1}=1$, that means $\tau(v)=v$, so
$v=a\in F^{\times}$. Therefore the critical relations are $z_{1}=y_{1}$,
$z_{2}=\tau^{2}(v)v^{-1}\sigma(y_{2})$, and there exists $a\in F^{\times}$
such that $g^{\prime}=ag$.

Assume $\sigma=\tau$, i.e. $y_{1}=\tau(v)v^{-1}\tau(y_{1})$. Then there is
$a\in F^{\times}$, $v=ay_{1}^{-1}$ and
$z_{2}=\tau^{2}(y_{1}^{-1})y_{1}\tau(y_{2})$, $g^{\prime}=a\tau
g\tau^{-1}L(y_{1})^{-1}$.

Assume $\sigma=\tau^{2}$, i.e. the critical relations are
$y_{1}=\tau^{2}(v)v^{-1}\tau^{2}(y_{1})$, which is the same as
$\tau(vy_{1})=vy_{1}$, i.e. $a=vy_{1}\in F^{\times}$, and
$g^{\prime}=\tau^{2}g\tau^{-2}L(v)$. Therefore the critical relations are
$z_{1}=y_{1}$, $z_{2}=\tau^{2}(v)v^{-1}\tau^{2}(y_{2})$ and
$g^{\prime}=a\tau^{2}g\tau^{-2}L(y_{1})^{-1}$.
(12) The critical relations are $z_{1}=\tau(v)v^{-1}\sigma(y_{1})$,
$z_{2}=\tau^{2}(v)v^{-1}\sigma(y_{2})$ and $g^{\prime}=\sigma
g\sigma^{-1}L(v)$, which imply $z_{1}=y_{1}$ by the definition of $M$.

Assume $\sigma=id$. This yields $\tau(v)v^{-1}=1$, that means $\tau(v)=v$, so
$v=a\in F^{\times}$. Therefore the critical relations are $z_{1}=y_{1}$,
$z_{2}=\tau^{2}(v)v^{-1}\sigma(y_{2})$, and there exists $a\in F^{\times}$
such that $g^{\prime}=ag$.

Assume $\sigma=\tau$, i.e. $y_{1}=\tau(v)v^{-1}\tau(y_{1})$. Then there is
$a\in F^{\times}$, $v=ay_{1}^{-1}$ and
$z_{2}=\tau^{2}(y_{1}^{-1})y_{1}\sigma(y_{2})$, $g^{\prime}=a\tau
g\tau^{-1}L(y_{1})^{-1}$.

Assume $\sigma=\tau^{2}$, i.e. the critical relations are
$y_{1}=\tau^{2}(v)v^{-1}\tau^{2}(y_{1})$, which is the same as
$\tau(vy_{1})=vy_{1}$, i.e. $a=vy_{1}\in F^{\times}$, and
$g^{\prime}=\tau^{2}g\tau^{-2}L(v)$. Therefore the critical relations are
$z_{1}=y_{1}$, $z_{2}=\tau^{2}(v)v^{-1}\sigma(y_{2})$ and
$g^{\prime}=a\tau^{2}g\tau^{-2}L(y_{1})^{-1}$.
(13) For $y_{2},z_{2}\in M$ and nonzero $y_{1},z_{1}\in K\setminus K^{\times}$
such that $N_{A_{0}}(y_{1}t+y_{2}t^{2})\not=0$ and $g,g^{\prime}\in{\rm
Gl(V)}$, we have $K^{(L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(L(z_{1})\tau+L(z_{2})\tau^{2},g^{\prime})}$ if and only if there are
$u,v\in K^{\times}$, $g^{\prime}=L(u^{-1})\sigma g\sigma^{-1}L(uv)$ and
$z_{1}=\tau(u)\tau(v)v^{-1}\sigma(y_{1}),$
$z_{2}=\tau^{2}(u)\tau^{2}(v)v^{-1}\sigma(y_{2}).$ This implies that
$N(z_{1})=N(u)N(y_{1})$ and $N(z_{2})=N(u)N(y_{2})$. If $N(u)=1$ we get
$z_{2}=y_{2}$ since $y_{2},z_{2}\in M$.
(14) $K\cong K^{(L(y_{1})\tau+L(y_{2})\tau^{2},g)}$ For $y_{1},z_{1}\in M$ and
nonzero $y_{2},z_{2}\in K\setminus K^{\times}$ and $g,g^{\prime}\in{\rm
Gl(V)}$, we have $K^{(L(y_{1})\tau+L(y_{2})\tau^{2},g)}\cong
K^{(L(z_{1})\tau+L(z_{2})\tau^{2},g^{\prime})}$ if and only if there are
$u,v\in K^{\times}$, $g^{\prime}=L(u^{-1})\sigma g\sigma^{-1}L(uv)$ and
$z_{1}=\tau(u)\tau(v)v^{-1}\sigma(y_{1}),$
$z_{2}=\tau^{2}(u)\tau^{2}(v)v^{-1}\sigma(y_{2}).$ This implies that
$N(z_{1})=N(u)N(y_{1})$ and $N(z_{2})=N(u)N(y_{2})$. If $N(u)=1$ we get
$z_{2}=y_{2}$ since $y_{1},z_{1}\in M$.
(15), (16), (17): The critical relations are $g^{\prime}=L(u^{-1})\sigma
g\sigma^{-1}L(uv)$, $z_{0}=u\sigma(y_{0}),$ and
$z_{i}=\tau^{i}(u)\tau^{i}(v)v^{-1}\sigma(y_{i})$ for all $i$, and cannot be
further simplified. If $N(u)=1$ we get $z_{2}=y_{2}$ since $y_{2},z_{2}\in M$
in (17).
(18) Here, the critical relations reduce to

$L(w_{0})+L(w_{1})\tau+L(w_{2})\tau^{2}=L(u)\sigma(L(x_{0})+L(x_{1})\tau+L(x_{2})\tau^{2})\sigma^{-1}L(uv)$
$=L(\sigma(x_{0})v)+L(\sigma(x_{1})\tau(v)\tau(u)u^{-1})\tau+L(\sigma(x_{2})\tau^{2}(v)\tau^{2}(u)u^{-1})\tau^{2}.$

By Corollary 2.3, this is equivalent to
$w_{0}=\sigma(x_{0})v,w_{1}=\sigma(x_{1})\tau(v)\tau(u)u^{-1},w_{2}=\sigma(x_{2})\tau^{2}(v)\tau^{2}(u)u^{-1}.$
∎

_Acknowledgments:_ This paper was written while the author was a visitor at
the University of Ottawa. She acknowledges support for her stay from the
Centre de Recherches Mathématiques for giving a colloquium talk, and from
Monica Nevins’ NSERC Discovery Grant RGPIN-2020-05020. She would like to thank
the Department of Mathematics and Statistics for its hospitality and the
congenial atmosphere, and in particular M. Nevins for lots of fruitful
discussions.

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