A classification of $C_{p^n}$-Tambara fields
Noah Wisdom

Two-pole structure of the h_1(1415) axial-vector meson:
  resolving mass discrepancy
    Hyun-Chul Kim
    September 9, 2024
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§ ABSTRACT 
	Tambara functors arise in equivariant homotopy theory as the structure adherent to the homotopy groups of a coherently commutative equivariant ring spectrum. We show that if k is a field-like C_p^n-Tambara functor, then k is the coinduction of a field-like C_p^s-Tambara functor ℓ such that ℓ(C_p^s/e) is a field. If this field has characteristic other than p, we observe that ℓ must be a fixed-point Tambara functor, and if the characteristic is p, we determine all possible forms of ℓ through an analysis of the behavior of the Frobenius endomorphism and an application of Artin-Schreier theory.





§ INTRODUCTION


For G a finite group, G-Tambara functors are the basic objects of study in equivariant algebra. They arise in homotopy theory as the natural structure adherent to the homotopy groups of a G-𝔼_∞ ring spectrum, though they additionally arise through many important situations in commutative algebra. For example any finite Galois field extension gives rise to a Gal-Tambara functor, and the representation rings of G and its subgroups naturally have the structure of a Tambara functor.

Roughly speaking, the notion of a G-Tambara functor is obtained by abstracting the notion of a Galois extension with Galois group G. More precisely, in this setting, one has intermediate fields for each subgroup H ⊂ G which have residual Weyl group W_G H action, contravariant inclusions between intermediate fields, as well as covariant transfer and norm maps between intermediate fields, all satisfying formulae relating various compositions. In a G-Tambara functor, we ask merely for rings k(G/H) for each subgroup of G, and do not require that restriction maps are inclusions. Here we still have transfers, norms, and Weyl group actions, whose compositions satisfy similar formulae. A morphism of G-Tambara functors is a collection of ring maps, one for each level G/H, which commute with restrictions, norms, transfers, and Weyl group actions.

While G-Tambara functors are the equivariant algebra analogues to rings, Nakaoka <cit.> <cit.> has defined field-like Tambara functors as those nonzero k for which every morphism k →ℓ with ℓ≠ 0 is monic. In particular, Nakaoka defines an ideal of a Tambara functor and shows that every Nakaoka ideal is obtained as the collection of kernels at each level of a map of G-Tambara functors. Next, Nakaoka observes <cit.> that k is field-like if and only if k(G/e) has no nontrivial G-invariant ideals and all restriction maps in k are injective. Additionally, upcoming work of Schuchardt, Spitz, and the author <cit.> classify the algebraically closed (or Nullstellensatzian) fields in Tambara functors: they are precisely the coinductions of algebraically closed fields.

Fields play an important role in homotopy theory and higher algebra; the rings _p are among the most fundamental objects, viewed as 𝔼_∞-ring spectra via the Eilenberg-MacLane construction. While this construction makes sense for any discrete ring, the most powerful computational tools of this form are usually obtained by feeding in a field. In equivariant homotopy theory, there is a similar Eilenberg-MacLane construction, although in the literature, computations are typically carried out with respect to the constant Tambara functors associated to fields (or the initial Tambara functor). These are indeed field-like Tambara functors, although they do not have the property that all of their Mackey functor modules are free! On the other hand, there are many other Tambara fields, for which there are relatively few computations in the literature, which do have the property that all of their Mackey functor modules are free (namely those which are coinduced from fields). We hope that the results of this article will serve as a source of inspiration for equivariant computations. For example, we pose the following question: what are the RO(C_p^n)-graded homotopy groups of all C_p^n-Tambara fields?

We aim to give a complete classification of the field-like C_p^n-Tambara functors, for C_p^n the cyclic group of order p^n. The impetus of this work is the following observation of David Chan and Ben Spitz <cit.>. They showed that if k is field-like, then k(G/e) is a product of copies of a field  permuted transitively by the G-action. Despite the fact that this may be deduced relatively quickly from Nakaoka's results, it suggests that an enormous amount of structure on a Tambara functor is forced by the field-like condition. To capture the special case of the Chan-Spitz result for which k(G/e) is a field, we introduce the following definition.


Let k be a Tambara functor. If k(G/e) ≅ Fun(G/H,R) for some H-ring R and proper subgroup H ⊂ G, we call k separated. Otherwise we call k pure. 


If a field-like Tambara functor k is separated, we may express this suggestively as k(G/e) ≅Coind_H^G, where Coind_H^G is the coinduction functor from H-rings to G-rings, right adjoint to the restriction morphism. A similar right adjoint exists on the level of Tambara functors, also called coinduction and written Coind_H^G. To reduce clutter, we introduce the notation Coind_i^n for the coinduction from C_p^i to C_p^n (and Res_i^n for the restriction from C_p^n to C_p^i).


	For C_p^n the cyclic order p^n group, if k is a field-like C_p^n-Tambara functor, then 
    k ≅Coind_i^n ℓ
 for some pure C_p^i-Tambara functor ℓ.


This reduces the classification problem of C_p^n-Tambara fields to those which are pure, ie have C_p^n/e level a field. If the characteristic of this field is prime to p, or the C_p^n-fixed point subfield is perfect, then the classification of such Tambara fields is straightforward.



	Suppose k is a pure field-like C_p^n-Tambara functor. If char(k(G/e)) ≠ p, or k(C_p^n/e)^C_p^n is perfect, then k is canonically isomorphic to the fixed-point Tambara functor associated to k(C_p^n/e), ie the restriction maps determine isomorphisms k(C_p^n/C_p^i) → k(C_p^n/e)^C_p^i.


In the case of characteristic p with k(C_p^n/e)^C_p^n nonperfect, it turns out we may still classify all possible structure. Roughly speaking, a pure field-like C_p^n-Tambara functor is obtained by choosing a descending collection of subrings of a field with C_p^n-action. The chief obstruction to an arbitrary collection of subrings forming a C_p^n-Tambara functor is that they must contain the appropriate norms and transfers. In particular, one may first show that all such subrings must be subfields which contain the image the C_p^n-fixed points under the nth iterate of the Frobenius endomorphism.

With this niceness condition, we describe how any pure C_p^n-Tambara field k of characteristic p may be constructed from suitably compatible pure C_p^n-1-Tambara and C_p-Tambara fields of characteristic p, respectively ℓ_t and ℓ_b (along with one additional minor piece of information). Recursively, this reduces the classification to pure C_p-Tambara fields of characteristic p. If the C_p action on the C_p/e level is trivial, this classification is straightforward, and if it is nontrivial, we utilize Artin-Schreier theory, culminating in the following proposition, which completes the classification.


	Every pure C_p-Tambara field of characteristic p is obtained by choosing a sub-Tambara-field of k(C_p/e) according to one of the following two situations:
 
	
  * Let C_p act trivially on , set k(C_p/e) =, and set k(C_p/C_p) to be any subfield of  containing the image of the Frobenius endomorphism.
	
  * Let →[x]/(x^p-x-α) be any Artin-Schreier field extension. Set k(C_p/e) = [x]/(x^p-x-α) (with C_p acting as the Galois group). If p is odd, choose k(C_p/C_p) any subfield of  containing both α and the image of  under the Frobenius endomorphism. If p is even, then k(C_p/C_p) must be .



Interestingly, the possible structure depends on whether p is odd or p = 2. In particular, there are "fewer" C_p^n-Tambara fields when p = 2 than when p is odd.

In section 2, we review the necessary background on field-like Tambara functors and coinduction. Section 3 provides the reduction of the classification problem to pure Tambara functors. Finally, section 4 explains how to construct any pure C_p^n-Tambara functor from pure C_p-Tambara functors, and classifies all pure C_p-Tambara functors.

The author would like to thank his advisor, Mike Hill, for many deep and insightful conversations. Additionally, the author thanks David Chan for sharing the proof of Proposition <ref>, due to David Chan and Ben Spitz. Finally, the author thanks Haochen Cheng for helpful conversations.



§ RECOLLECTIONS ON TAMBARA FUNCTORS


For a complete introduction to Tambara functors, see <cit.>. Recall that, for G a finite group, a G-Tambara functor k is roughly the following data:

	
  * Rings k(G/H) for each transitive G-set G/H. We say k(G/H) is in level G/H, and refer to k(G/e) (resp. k(G/G)) as the bottom level (resp. top).
	
  * Ring maps k(G/H) → k(G/K) for every morphism of G-sets G/K → G/H.
	
  * Multiplicative norm and additive transfer maps k(G/H) → k(G/K) for every morphism of G-sets G/H → G/K.


Note that the Weyl group W_H = N_H/H of H ⊂ G describes the automorphisms of the transitive G-set G/H, hence the rings k(G/H) all possesses Weyl group actions, which are intertwined by the restriction maps. The norm, transfer, and restriction maps are required to satisfy various formulae. One of these is the double coset formula, which we describe here under the assumption that G is abelian. For H ⊂ L, we have that the composition of the transfer T_L^H : k(G/H) → k(G/L) followed by restriction R_L^H : k(G/L) → k(G/H) is equal to the sum of the Weyl group orbits 
    R_L^H T_L^H = Σ_g ∈ L/H c_g
 where c_g denotes the action of g ∈ G → W_H on k(G/H). An analogous formula holds for the norm in place of the transfer, where the sum is replaced with a product.

Finally, given a Tambara functor k, we may identify it with the unique extension to a product-preserving functor from finite G-sets to rings; by product preserving, we mean k(G/H ⊔ G/K) ≅ k(G/H) × k(G/K). This perspective will be required for the discussion of coinduction below.


Let R be a ring with G-action. The fixed points Tambara functor is the G-Tambara functor R with R(G/H) = R^H. Noting that all restriction maps are inclusions, transfers and norms are uniquely defined as the appropriate sums (resp. products) of orbits via the double coset formula. The fixed point G-Tambara functor construction is functorial, and right adjoint to the functor k ↦ k(G/e) from G-Tambara functors to G-rings.



A nonzero G-Tambara functor k is called field-like, or a G-Tambara field, if every nonzero morphism with domain k is monic.


By this definition, any field-like Tambara functor k may be viewed as a subfunctor of the field-like Tambara functor k(G/e). This is because the adjunction unit k →k(G/e) is nonzero, hence monic (hence injective in all levels). By this fact, to specify a field-like Tambara field, it is enough to specify a subring of each level of a Tambara field R which collectively are appropriately closed under taking transfers, norms, and restrictions.


A G-Tambara functor k is field-like if and only if all restriction maps are injective and k(G/e) has no G-invariant ideals (recalling W_e = G).


Directly from this, we may prove the following result of David Chan and Ben Spitz. While this is an elementary consequence of the statement that k(G/e) has no G-invariant ideals (in fact, it is equivalent to it), it greatly illuminates the structure of Tambara fields.



	Let k be a field-like G-Tambara functor. Then k(G/e) is a product of copies of a field  permuted transitively by the G-action.



	Let m be a maximal ideal of k(G/e), and consider the G-set { gm | g ∈ G }. Since G acts transitively, it is isomorphic to G/H for some subgroup H ⊂ G. Now consider ∩_gH ∈ G/H gm. This is a G-invariant ideal, hence by <cit.> it must be the zero ideal. Writing = k(G/e)/gm (which does not depend on the choice of g), by the Chinese remainder theorem, k(G/e) ≅^|G/H|. Since G acts transitively on the gm, it transitively permutes the factors in the product.


This result suggests the following definition, with which we reinterpret Nakaoka's result.


A G-ring is field-like if it has no nontrivial G-invariant ideals. Equivalently, it is a product of fields permuted transitively by the G action.



A Tambara functor k is field-like if and only if all restriction maps are injective and k(G/e) is a field-like G-ring.


Without knowing Proposition <ref> or <cit.>, it is a priori possible for a field-like Tambara functor k with k(G/e) ≅/n to exist for some composite integer n. Fortunately, there is an intrinsic notion of characteristic of a G-Tambara functor, which by Proposition <ref> may be identified with the usual possibilities for characteristic of a field.


The characteristic of a Tambara functor k is the equivalence class determined by the following equivalence relation: k ∼ℓ if k ⊠ℓ≠ 0.



The characteristic of k may be identified with the characteristic of k(G/e).



Use the formula (k ⊠ℓ)(G/e) ≅ k(G/e) ⊗ℓ(G/e) and the fact that k(G/e) and ℓ(G/e) are finite products of fields.


There is likely interesting combinatorial structure captured by the box-product of field-like Tambara functors, although a more serious investigation falls outside the scope of this paper.

Finally, we review the coinduction functor. Given H ⊂ G, the coinduction Coind_H^G ℓ of an H-Tambara ℓ to a G-Tambara functor is obtained by precomposition with the restriction functor from finite G-sets to finite H-sets. This requires us to view ℓ as a functor on all finite G-sets, rather than merely the transitive ones. For G = C_p^n and ℓ a C_p^k-Tambara functor, we supply a pictoral description of the coinduction Coind_k^n ℓ below:

    ( Coind_k^n ℓ) ( C_p^n/C_p^n)    ≅ℓ(C_p^k/C_p^k) 
    ( Coind_k^n ℓ) ( C_p^n/C_p^n-1)    ≅ℓ(C_p^k/C_p^k)^× p
       ⋮
    ( Coind_k^n ℓ) ( C_p^n/C_p^k+1)    ≅ℓ(C_p^k/C_p^k)^× p^n-k-1
    ( Coind_k^n ℓ) ( C_p^n/C_p^k)    ≅ℓ(C_p^k/C_p^k)^× p^n-k
    ( Coind_k^n ℓ) ( C_p^n/C_p^k-1)    ≅ℓ(C_p^k/C_p^k-1)^× p^n-k
       ⋮
    ( Coind_k^n ℓ) ( C_p^n/e )    ≅ℓ(C_p^k/e)^× p^n-k.


One immediately observes using <cit.> that if ℓ is a field-like H-Tambara functor, then so is Coind_H^G ℓ. Coinduction is right adjoint to the restriction functor Res_H^G, which is given levelwise by precomposition with the coinduction functor from H-sets to G-sets <cit.>. Heuristically, one may view coinduction as "preserving the top level" and restriction as "preserving the bottom level". In particular, restriction does not in general preserve Tambara fields, although we have the following.


Suppose k is a pure G-Tambara field. Then for any subgroup H ⊂ G, Res_H^G k is a pure H-Tambara field.


There is also a coinduction functor from H-rings to G-rings, which is right adjoint to the restriction functor. It is given by R ↦ Fun(G/H,R), which we abbreviate by Coind_H^G R.



§ SEPARATED TAMBARA FIELDS


In this section we aim to reduce the classification of field-like C_p^n-Tambara functors to those whose bottom level C_p^n/e is a field. To describe Tambara fields of this form, we introduce the notion of a pure Tambara functor.


Let k be a Tambara functor. If k(G/e) ≅Coind_H^G R for some ring R and proper subgroup H ⊂ G, we call k separated. Otherwise we call k pure. 




	Let R an H-ring. Then Coind_H^G R≅Coind_H^G R.



	Since coinduction is right adjoint to restriction and the fixed-point construction is right adjoint to the "bottom-level" functor, it suffices to prove that the left adjoints commute, ie for all G-Tambara functors k, we have 
    ( Res_H^G k ) (H/e) ≅Res_H^G (k(G/e)) .
 Now the left-hand side is defined as k ( Coind_H^G (H/e) ) ≅ k(G/e) with H acting through restriction of the G-action. This is precisely Res_H^G (k(G/e)), as desired.



Let k a G-Tambara functor with k(G/e) ≅Coind_H^G R for some H-ring R, and suppose that the restriction map k(G/H) → k(G/e) is injective. Then k(G/H) ≅Coind_H^G S for some subring S ⊂ R.



	Let { x_gH} denote the set of idempotents corresponding to projection on the each factor R in level G/e. Note that this set is isomorphic to G/H. The double coset formula implies that the composition of the norm map from the bottom level G/e to level G/H with the restriction map to the bottom level sends each x_gH to itself (the product over the H-orbits). Therefore k(G/H) contains a sub-G-set of idempotents isomorphic to G/H, which induce the desired isomorphism.




Suppose k is a field-like G-Tambara functor and k(G/e) ≅Coind_H^G for some H-field . Then whenever L ⊂ H, k(G/L) ≅Coind_H^G R for some subring R of .



The restriction map k(G/H) → k(G/e) factors through k(G/L), hence the sub-G-set of idempotents of k(G/H) isomorphic to G/H is also contained in k(G/L)




	Suppose G is abelian, k is any G-Tambara functor such that k(G/H) is a product of copies of some ring R permuted freely and transitively by the Weyl group G/H, and L is a subgroup of G containing H such that the restriction k(G/L) → k(G/H) is injective. Then the restriction map has image k(G/H)^L.



	Choose an idempotent x ∈ k(G/H) corresponding to projection onto a factor R and choose r ∈ R arbitrary. The double coset formula implies that transferring rx up to k(G/L) and restricting the resulting element down to k(G/H) results in 
    r ( Σ_g ∈ L/H gx ) .
 Repeating this process through all choices of x and r ∈ R, we observe that the image of the restriction contains a collection of copies of R, embedded in k(G/H) via the diagonal embedding R → R^× L/H followed by any of the |G/L| inclusions R^× L/H↪ R^× G/H. Therefore the subring generated by the image is precisely the L-fixed points of R^× G/H.


The previous two lemmas show that any field-like G-Tambara functor "looks like" a coinduced one in any level G/L such that L either contains or is contained in some fixed subgroup H ⊂ G. So, we can only deduce that field-like Tambara functors are always coinduced for families of abelian groups for which the subgroup lattice is a well-ordered set. This is why we only obtain a classification of fields for groups C_p^n. The extent to which field-like Tambara functors are coinduced from pure ones for more general G remains open (though likely not difficult).



	If k is a field-like C_p^n-Tambara functor, then k ≅Coind_s^n ℓ for some pure C_p^s-Tambara functor ℓ.



	By Proposition <ref>, k(C_p/e) ≅Coind_s^n for some C_p^s-field . Composing the canonical map to the fixed point Tambara functor of the C_p^n/e level with the isomorphism of Lemma <ref> supplies a map k →Coind_s^n which is manifestly an isomorphism in level C_p^n/e.
	
	As rings, set ℓ(C_p^s/C_p^j) to be the subring R_j of  appearing in Corollary <ref>, and identify k(C_p^n/C_p^j) with Coind_s^n ℓ(C_p^s/C_p^j). The C_p^s-equivariant restriction maps for ℓ are obtained as the restriction of the restriction maps for k to the eC_p^s factor (the proof of Corollary <ref> shows that this is well-defined). The norm and transfer maps are defined similarly, observing that the double coset formula along with injectivity of the restriction maps imply that the restriction of the norm (resp. transfer) in k(C_p^n/C_p^j) to the eC_p^s factor lands in the eC_p^s factor, for j ≤ s. The exponential and double coset formulae for k then become the double coset formulae for ℓ. 
	
	We may alternatively construct ℓ as follows. Note that Res_s^n k has an action of C_p^n/C_p^s arising from the free and transitive permutation of the C_p^s-orbits of the C_p^s-sets 
    Res_s^n ( Coind_s^n C_p^s/C_p^k)
 which corresponds in each level to permuting the |C_p^n/C_p^s| factors ℓ(C_p^s/C_p^j) of k(C_p^n/C_p^j). We define ℓ as the subfunctor of Res_s^n k formed by the C_p^n/C_p^s-fixed points of this action.
	
	Now ℓ is a pure field-like C_p^s-Tambara functor, and we may coinduce the canonical map 
    ℓ→ℓ(C_p^s/e)
 to 
    Coind_s^n ℓ→Coind_s^n ℓ(C_p^s/e)≅Coind_s^n
 Finally, the image of 
    k(C_p^n/C_p^i) →Coind_s^n (C_p^n/C_p^i)
 is precisely the image of Coind_s^n ℓ(C_p^n/C_p^i); when i ≤ k this is by construction of ℓ, and when i ≥ k this is by Lemma <ref>. Since k and Coind_s^n ℓ are both field-like, they are naturally isomorphic to their images in Coind_s^n, hence to each other.




§ PURE TAMBARA FIELDS


In this section we aim to classify the pure Tambara fields. In many cases the double coset formula forces pure Tambara fields to be isomorphic to fixed-point Tambara functors; we start by collecting these results.



Suppose k is a pure C_p^n-Tambara functor of characteristic different from p. Then the canonical map k →k(C_p^n/e) is an isomorphism.



Consider the restriction of the transfer map k(C_p^n/e) → k(C_p^n/C_p^s) to the C_p^s-fixed points. The double coset formula implies that postcomposing this map with the restriction k(C_p^n/C_p^s) → k(C_p^n/e) is multiplication by p^s, which is a unit in k(C_p^n/e) by assumption. Therefore the restriction has image k(C_p^n/e)^C_p^s. Since it is injective by Nakaoka's theorem, it is an isomorphism k(C_p^n/C_p^s) ≅ k(C_p^n/e)^C_p^s. This is precisely the statement that k →k(C_p^n/e) is an isomorphism.



The category of field-like C_p^n-Tambara functors of characteristic other than p is adjointly equivalent to the category of field-like C_p^n-rings of characteristic other than p.



By Proposition <ref> and Theorem <ref>, the functor R ↦R is an inverse adjoint equivalence to the functor k ↦ k(G/e).


In general, we conjecture that the above corollary is true for any finite group G, so long as the characteristic does not divide the order of the group. It seems more likely that this is at least true for abelian groups G.


Let k be a field-like C_p^n-Tambara functor of characteristic different from p. Then any morphism of field-like C_p^n-Tambara functors ℓ→ k which is an isomorphism on the bottom level C_p^n/e is an isomorphism of Tambara functors.


This corollary may be of homotopical use. Namely, it heuristically says that the C_p^n/e level homotopy group functor is conservative on C_p^n-𝔼_∞-ring spectra whose homotopy groups are appropriately built out of field-like Tambara functors of characteristic other than p.

We will see later that these corollaries fail in characteristic p. Recall that Artin's lemma states that if a finite group G acts on a field , then the inclusion of G-fixed points ^G → is a Galois extension. The Galois group is the homomorphic image of G in Aut().



Suppose k is a pure C_p^n-Tambara functor such that k(C_p^n/e)^C_p^n is a perfect characteristic p field. Then the canonical map k →k(C_p^n/e) is an isomorphism.



As in the argument of Proposition <ref>, it suffices to show that each restriction map k(C_p^n/C_p^s) → k(C_p^n/e) has image k(C_p^n/e)^C_p^s. Since any Galois extension of a perfect field is perfect, our assumption ensures that each fixed-point field k(C_p^n/e)^C_p^s is perfect.

Now consider the restriction of the norm k(C_p^n/e) → k(C_p^n/C_p^s) to the C_p^s-fixed points. The double coset formula implies that postcomposing this map with the restriction k(C_p^n/C_p^s) → k(C_p^n/e)^C_p^s is x ↦ x^p^s, ie the s-fold iterate of the Frobenius map. Since k(C_p^n/e)^C_p^s is perfect, the Frobenius map is an isomorphism, so we observe that the restriction map is surjective, as desired.


Combining Proposition <ref> with Proposition <ref>, we obtain Theorem <ref>. Next, we analyze what happens when k is a pure Tambara field with k(C_p^n/e)^C_p^n a possibly non-perfect characteristic p field.


Let k be a pure C_p^n-Tambara functor of characteristic p. Writing ϕ for the Frobenius endomorphism, we call the subfield ϕ^n(k(C_p^n/e)^C_p^n) of k(C_p^n/e) the lower bound field of k.



Suppose k is a pure C_p^n-Tambara functor of characteristic p. Then each k(C_p^n/C_p^s), viewed as a subring of k(C_p^n/e), is an intermediate field of the extension ϕ^n ( k(C_p^n/e)^C_p^n) ↪ k(C_p^n)



By the double coset formula, the lower bound field of k is contained in the image of the composition of the norm map k(C_p^n/e) → k(C_p^n/C_p^n) with the restriction k(C_p^n/C_p^n) → k(C_p^n/e). In particular, it is contained in the image of all restriction maps. Therefore each k(C_p^n/C_p^s) is a subring of k(C_p^n/e) (via the restriction map) containing the lower bound field.

To show each k(C_p^n/C_p^s) is a field, it suffices to show each element has an inverse. Note that k(C_p^n/e) is algebraic over the lower bound field, because it is a Galois extension of its C_p^n-fixed point subfield by Artin's lemma and any characteristic p field is algebraic over the image of an iterate of the Frobenius endomorphism.

Letting x ∈ k(C_p^n/C_p^s), we see that x is a root of some polynomial over the lower bound field. In particular, the subring of k(C_p^n/C_p^s) generated by x and the lower bound field is a finite-dimensional vector space over the lower bound field, hence is Artinian. Since it is a subring of a field, it is an integral domain, hence a field. Thus x has an inverse in k(C_p^n/C_p^s).


Let k be a pure C_p^n-Tambara functor of characteristic p. Then we may construct a C_p^n-1-Tambara functor ℓ_t which captures the "top piece" of k as follows. Observe that for s ≥ 1 each k(C_p^n/C_p^s) has a C_p^n-1 action with kernel C_p^s-1 (namely, regard the Weyl group as a quotient of C_p^n). First, set ℓ_t(C_p^n-1/C_p^s-1) = k(C_p^n/C_p^s) for 1 ≤ s ≤ n. Next, define the restriction maps for ℓ_t via the restriction maps for k. Since the restriction maps for k are appropriately equivariant, so are those for ℓ_t.

Finally, define the norm and transfer maps for ℓ_t via the norm and transfer maps for k with the appropriate domain and codomain. To check that ℓ_t is a C_p^n-1-Tambara functor, it suffices to check that the appropriate double coset and exponential formulae are satisfied. In fact, we may do this in a universal example. Since we have already defined norms and transfers on ℓ_t, via the map k →k(C_p^n/e) it suffices to check that our construction produces a C_p^n-1-Tambara functor when applied to a fixed-point Tambara field . This is clear, however, as our construction produces the fixed-point C_p^n-1-Tambara functor ^C_p^n-1.

On the other hand, we may extract a C_p-Tambara field ℓ_b from k which recovers the "bottom piece" of k by ℓ_b := Res_1^n k. Unwinding definitions, we have ℓ_b(C_p/e) = Res_1^n k(C_p^n/e) and ℓ_b(C_p/C_p) = Res_0^n-1 k(C_p^n/C_p), with restriction, norm, and transfer for k giving the restriction, norm, and transfer maps for ℓ_b.


Every pure C_p^b-Tambara field k of characteristic p is obtained from the following: 

	
  * a choice of pure C_p^n-1-Tambara field ℓ_t of characteristic p
	
  * a choice of C_p^n-field = k(C_p^n/e)
	
  * a choice of pure C_p-Tambara field ℓ_b of characteristic p.

These choices must satisfy the following compatibility criteria:

	
  * ℓ_b(C_p/C_p) = Res_0^n-1ℓ_t(C_p^n-1/e)
	
  * ℓ_b(C_p/e) = Res_1^n
	
  * The ring map 
    ℓ_t(C_p^n-1/e) ≅ℓ_b(C_p/C_p) →ℓ_b(C_p/e) ≅
 is C_p^n-1-equivariant.




Given ℓ_b, ℓ_t, and k(C_p^n/e) as above, we define a C_p^n-Tambara functor k as the following subfunctor of the fixed-point C_p^n-Tambara functor . Set k(C_p^n/e) = and k(C_p^n/C_p^s) = ℓ_t(C_p^n-1/C_p^s-1) for s ≥ 1. The restrictions, norms, and transfers which do not factor nontrivially through C_p^n/C_p are well-defined (in the sense that their codomain contains their image) because they are well-defined for ℓ_t and ℓ_b respectively. The remaining restrictions, norms, and transfers are well-defined because they are compositions of well-defined restrictions, norms, and transfers, respectively.


This recursively reduces the classification of pure C_p^n-Tambara fields of characteristic p to pure C_p-Tambara fields of  characteristic p. Let k be such a C_p-Tambara functor. If C_p acts trivially on k(C_p/e), then the composition of the norm map with the restriction may be identified with the Frobenius endomorphism, and the transfer map is zero. Thus k(C_p/C_p) may be any subfield of k(C_p/e)^C_p containing the image of the Frobenius endomorphism.


We may form a C_p-Tambara functor of the above type as follows. First, consider the fixed-point Tambara functor associated to the trivial C_p action on _p(t). We may form a sub-Tambara functor with the same bottom level C_p/e, but top level equal to the image of the Frobenius endomorphism _p(t^p). The inclusion of this sub-functor provides an example of a morphism between Tambara fields which is an isomorphism on the bottom level, but is not an isomorphism.



We may also form a C_p-Tambara functor with nontrivial C_p-action on the bottom level. Let k(C_p/e) = _p(t)[x]/(x^p-x-t) with C_p acting as the Galois group over _p(t). The transfer of x is 0 (or -1 for p=2), and the norm of x is -t. So k(C_p/C_p) must not only contain the image of the Frobenius endomorphism on _p(t), but t as well. Consequently k(C_p/C_p) must be _p(t) (since it must be a subfield of the fixed points). On the other hand, we could have chosen k(C_p/e) = _p(t)[x]/(x^p-x-1), in which case we could have set k(C_p/C_p) = _p(t^p) (for p > 2). For p = 2, the transfer of tx is -t, so we must have k(C_p/C_p) = _p(t).


Suppose k is a pure C_p-Tambara field of characteristic p with C_p acting nontrivially on k(C_p/e). Then k(C_p/C_p) may be any subfield of k(C_p/e)^C_p which contains the image of the Frobenius endomorphism, as well as the image of the norm and transfer from k(C_p/C_p). 

This observation may be further refined by studying the possible norm and transfer maps. First, note that k(C_p/e) is a Galois extension of k(C_p/e)^C_p with Galois group C_p. Artin-Schreier theory <cit.> implies that k(C_p/e) is the splitting field of an Artin-Schreier polynomial, which has the form x^p-x-α, where α is fixed by C_p. The norm of any root of this polynomial is the constant term - α, and the transfer of any root is either 0 when p ≠ 2 or -1 for p = 2.

The arbitrary element of k(C_p/e) is a k(C_p/e)^C_p-linear combination of powers of some fixed root of an Artin-Schreier polynomial. Now observe that the norm of a sum of elements may be expressed as a polynomial combination of the norms of the summands and transfers of products of the summands <cit.>. Hence any subfield containing -α, the lower bound field (the image of k(C_p/e)^C_p under the Frobenius endomorphism), and the image of k(C_p/e) under the transfer contains the norm of every element in k(C_p/e). 

Next, since the transfer is additive, it suffices to determine the transfer of an arbitrary k(C_p/e)^C_p-multiple of a root of an Artin-Schreier polynomial. Restricting down to k(C_p/e), we get the sum over the C_p-orbits; since any b ∈ k(C_p/e)^C_p is fixed, we may pull b out of the sum. If p is odd, we obtain zero, but if p = 2, we obtain -b. In this case the image of the transfer is k(C_p/e)^C_p.



	Every pure C_p-Tambara field of characteristic p is obtained by choosing a sub-Tambara-field of k(C_p/e) according to one of the following two situations:
 
	
  * Let C_p act trivially on , set k(C_p/e) =, and set k(C_p/C_p) to be any subfield of  containing the lower bound field.
	
  * Let →[x]/(x^p-x-α) be any Artin-Schreier field extension. Set k(C_p/e) = [x]/(x^p-x-α) (with C_p acting as the Galois group). If p is odd, choose k(C_p/C_p) any subfield of  containing both α and the lower bound field. If p is even, then k(C_p/C_p) must be .



This concludes the classification of field-like C_p^n-Tambara fields.

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