aainstitutetext: Departamento de Ingeniería Mecánica, Universidad de
Guanajuato, Carretera Salamanca-Valle de Santiago Km 3.5+1.8 Comunidad de Palo
Blanco, Salamanca, Mexico.bbinstitutetext: Department of Mechanical
Engineering, Virginia Tech, Blacksburg, VA 24061, USA.ccinstitutetext:
Departamento de Física, Universidad de Guanajuato, Loma del Bosque No. 103
Col. Lomas del Campestre C.P 37150 Leon, Guanajuato, Mexico.

# Galois groups of uplifted de Sitter vacua

Cesar Damian c and Oscar Loaiza-Brito

###### Abstract

We compute the Galois group of a polynomial whose roots are determined by the
critical points of a scalar potential in type IIB compactifications. We focus
our study on certain perturbative models where it is feasible to construct a
de Sitter vacuum within the effective theory by introducing non-geometric
fluxes, D-branes or non-BPS states. Our findings clearly show that all de
Sitter vacua derived from lifting AdS stable vacua are associated with an
unsolvable Galois group. This suggests a deeper connection between the
fundamental principles of Galois theory and its applications in the
construction of dS vacua.

###### Keywords:

Galois theory, dS vacua, Swampland

##  1 Introduction

The Swampland program is a research program in string theory that aims to
identify which effective quantum field theories can be consistently embedded
into a theory of quantum gravity, such as string theory, and which cannot. It
is motivated by the observation that many seemingly consistent effective
quantum field theories appear to be incompatible with some expected and
verified principles of quantum gravity, which is studied by general scenarios
in string theory (generally involving black hole physics). The program aims to
identify and understand these inconsistencies and use them to constrain the
space of quantum field theories with an ultraviolet completion in string
theory Palti:2019pca ; vanBeest:2021lhn ; Grana:2021zvf ; Agmon:2022thq .

In the last few years, important insights into string theory and quantum
gravity have been identified by proposing several different conjectures that
appear to be correlated nonetheless. This has stimulated research into the
connections between string theory, cosmology, and other areas of physics. One
of these is the so-called Refined de Sitter Conjecture (RdSC)Garg:2018reu ;
Ooguri:2018wrx , which states that the existence of stable de Sitter vacua is
excluded at least in asymptotic regions of the moduli space. However, unlike
well-established conjectures such as the weak gravity or the absence of global
symmetries, the RdSC is at the moment a proposal based mainly on the enormous
difficulty to construct stable de Sitter vacua from string compactifications
rather than the consequence of a more fundamental constraint in a quantum
gravity theory, although there are some strong grounds to consider its
validity Andriot:2018mav ; Blumenhagen:2019kqm ; Seo:2019mfk ;
Brandenberger:2020oav ; Damian2023some .

One of the most useful techniques to construct AdS vacuum is to uplift some
direction which is previously stabilized at some non-positive value, although
it is also possible to uplift a runaway direction. The most well-known
scenarios are the KKLT and the LVS, where the stability of the de Sitter
vacuum has been extensively studied in the last twenty years (for a recent
overview see Leontaris:2023obe ). More simple scenarios, such as toroidal
compactifications in the presence of fluxes, also exhibit the presence of
apparently stable de Sitter minima in some directions of the scalar potential
which are compromised by topological constraints or by the inclusion of exotic
fluxes Damian:2013dq ; Damian:2013dwa ; Blumenhagen:2015kja ; Damian:2018tlf ;
Plauschinn:2018wbo ; Andriot:2019wrs ; Shukla:2019wfo ; Shukla:2022srx .
However, these scenarios offer the possibility to study a relationship between
the presence of de Sitter minima and algebraic properties of the scalar
potential and its roots since in these cases they are polynomials of order
cubic at most on the moduli. In this context, we are interested in searching
for some inherent algebraic characteristics of a polynomial scalar potential
related to scenarios where one or more directions with some negative or null
minima are uplifted to a positive one.

For that, we examine four different scenarios in which it is possible to
uplift some directions to stable de Sitter minima by adding extra structure
such as non-geometric fluxes, non-BPS states, or by relaying in a perturbative
polynomial truncation of the superpotential related to a Calabi-Yau manifold
with large complex structure as reported in Coudarchet:2022fcl . In all of
these cases, the existence of such minima has been numerically demonstrated.

In this work, we show that there is a common algebraic characteristic
associated with these scenarios, specifically that the Galois group of the
polynomial scalar potential is unsolvable when uplifted to a de Sitter minima,
but solvable otherwise. We believe that this indicates a deeper relationship
between the algebraic properties of the scalar potential and the existence of
apparently stable de Sitter vacua. Since all of these vacua are related to
scenarios for which their ultraviolet completion is not clear, one could say
that a de Sitter emphasis vacuum constructed perturbatively possesses a scalar
potential with an unsolvable Galois group.

Our work is organized as follows: In section 2 we provide some basics on the
Galois theory by covering the essentials to construct the Galois group.
Sections 3 to 6 are focused on the study of different cases in which we can
uplift some directions to a positively valued minimum. In all of them we find
that previous to the uplifting, the associated Galois group is solvable but
becomes unsolvable once we manage to uplift the minimum to a dS one. We show
our conclusions and final comments in Section 7, while in the Appendix we
present some techniques that could be useful to the reader.

##  2 A brief overview on Galois group

Our goal in this work is to compute the Galois group for different scenarios
involving a polynomial scalar potential and to look for a relation between the
different stable vacua we can construct and the corresponding Galois group. To
that end, we shall start with a very brief overview of Galois Theory and its
importance in the existence of different vacua.

###  2.1 The Galois group

The Galois group of a polynomial is a fundamental concept in algebraic number
theory as well as algebraic geometry. Algebraically the Galois group is the
group of automorphisms of the field generated by the extension of the rational
field by including the roots of the polynomial, such that the field operations
are preserved. This group provides important information on the structure and
relation of the roots of the polynomial.

The field extension is a larger field that usually contains as the base field
the rationals. For instance, the biggest field extension over the reals is the
field of complex numbers. In the same manner, the field of algebraic numbers
is an extension of rational numbers and encodes information regarding the
symmetries of the field extension in its associated Galois group. The
computation of the Galios group of a field extension is carried out by finding
all the automorphisms of the extension that left fixed the base field and
indeed is a bijective map that preserves the field operations.

For example, for $n\in\mathbf{Z}^{+}$, the field extension generated by the
polynomial $p(x)=x^{2}-n$ over the field of rational numbers contains the
square root of $n$. The automorphisms, $\sigma$ that fix the rational numbers
must take $\sqrt{n}$ to another root of the polynomial. In this case,
$-\sqrt{n}$. Thus, the automorphism that takes $\sigma(\sqrt{n})$ to
$-\sqrt{n}$ is one element of the Galois group.

Since the computation of the Galois group could be a non-familiar calculation
in physics, we give a short and very summarized set of steps we have used to
compute it, corresponding to an implementation of the Stauduhar method to
determine the Galois group awtrey2017determining ; wildsmithcontribution :

  1. 1.

Once we have constructed the associated polynomial to some moduli field from
the scalar potential, we check whether the polynomial is irreducible over its
base field. If it is reducible, then we can factor it into irreducible factors
and compute the Galois group of each irreducible factor separately.

  2. 2.

Next, we need to find all the roots of every irreducible polynomial, which are
the elements of the splitting field. We can use algebraic or numerical methods
to find the roots, and the roots are ordered in an arbitrary manner.

  3. 3.

Let us say that $\text{Gal}~{}p$ is the Galois group of the polynomial $p(x)$
with respect to the initial arbitrary ordering of the corresponding roots. One
way to figure out what $\text{Gal}~{}p$ is, consists on identify possible
subgroups $M$. To determine if $M$ is a subgroup of $\Gamma$ one can follow a
set of steps as described in Ref. stauduhar1973determination ; GEISSLER2000653
. Broadly speaking one requires to compute the resolvent of the polynomial.
The resolvent is a monic polynomial $p$ with integer coefficients to which one
can relate some trimmed invariants, as the discriminant (to be defined later
on). These invariants are tested for integer values: if no integer values are
found then it is concluded that $\Gamma$ are not contained in any conjugates
of $M$, and similar resolvents are computed for other conjugacy classes of the
maximal transitive subgroup of Sn. On the other hand, if an integer root is
found for the coset representative $\pi_{i}$, this is used to re-order the
roots, and then $\text{Gal}~{}p$ is a subgroup $\pi p\pi^{-1}$.

  4. 4.

This procedure is repeated over a so-called decision tree to check all the
possibilities to uniquely determine the Galois group wildsmithcontribution .
In the cases we shall explore, we use some results directly from those
reported in references wildsmithcontribution ; SUTHERLAND201573 .

A Galois group is unsolvable if it is not a solvable group. A solvable group
is a group that can be solved using a sequence of abelian groups and their
extensions. The fundamental theorem of Galois theory implies that the
solvability of the Galois group is directly related to the fact that a
polynomial can be solved for radials, namely, whether the roots of the
polynomial can be expressed by elementary operations of a field, such as
multiplication, division, and square roots. Thus, the Abel-Ruffini theorem,
states that polynomials of degree five and higher are not solvable for
radicals, always that its Galois group is not solvable stewart2022galois .In
all the cases we shall study, the polynomial related to dS vacua are of degree
greater than 5.

Currently, it is possible to find several computational tools as well as
algorithms to compute the Galois theory. Some common tools employed are
specialized software such as Mathematica, Sagemath, and Magma which have
built-in functions for computing Galois groups, which allows one to determine
the Galois group by determining the action of the permutation group acting on
the roots 111 Since we are mainly interested in transitive groups and we shall
relate the groups given in the decision tree reported in wildsmithcontribution
, with the common notation given for transitive groups of degree less than 32,
as $nTm$ where $n$ is the degree of the group and $m$ is the number of the
group in the classification transitivegroups  and take its corresponding group
alias..

In the subsequent sections, we will explore four different scenarios where the
uplifting mechanism from a stable AdS vacuum to a dS one is constructed by the
addition of anti $D3$-branes, non-geometric fluxes, and non-BPS states.
Through these examples, we shall conclude that all such uplifting scenarios
are characterized by a lack of analyticity in the process and in consequence
are linked to an unsolvable Galois group.

##  3 Example 1: A model without complex structure

Let us consider a IIB string compactification on a CY with a single Kähler
modulus and without complex structure in the presence of a NS-NS flux $h$, a
RR flux $f$ and a non-geometric flux $q$ (model A in Blumenhagen:2015kja ).
This emulates a six-dimensional torus with a frozen complex structure. The
superpotential is given by

$\begin{split}W=i\tilde{f}+ihS+iqT\end{split}$ (1)

with Kähler moduli

$\begin{split}K=-3\log\left(T+\bar{T}\right)-\log\left(S+\bar{S}\right)\end{split}$
(2)

Hence, the scalar potential takes the form

$\begin{split}V=\frac{c^{2}h^{2}}{16s\tau^{3}}-\frac{ch\rho
q}{8s\tau^{3}}-\frac{\tilde{f}h}{8\tau^{3}}-\frac{\tilde{f}q}{8s\tau^{2}}+\frac{\tilde{f}^{2}}{16s\tau^{3}}-\frac{3hq}{8\tau^{2}}+\frac{h^{2}s}{16\tau^{3}}+\frac{\rho^{2}q^{2}}{16s\tau^{3}}-\frac{5q^{2}}{48s\tau}\end{split}$
(3)

with $S=s+ic$ and $T=\tau+i\rho$. In our next step, we shall compute the
Galois group related to the AdS minimum.

###  3.1 AdS free tachyon

The critical points of V are located at the zeros of the two polynomial
equations given by $\partial_{s,\tau}V=0$ with $\partial_{c,\rho}V$
identically zero at $c=\rho=0$. The zeros are located at

$\begin{split}\left(s,\tau\right)=\left[\left(-\frac{\tilde{f}}{2h},-\frac{3}{2}\frac{\tilde{f}}{q}\right),\left(-\frac{\tilde{f}}{h},-\frac{6}{5}\frac{\tilde{f}}{q}\right),\left(\frac{\tilde{f}}{8h},\frac{3}{8}\frac{\tilde{f}}{q}\right)\right]\end{split}$
(4)

where, as shown in Blumenhagen:2015kja , only the second solution is free of
tachyons. This vacuum is AdS with an energy value given by

$\begin{split}V_{\text{min}}=-\frac{5^{2}}{2^{3}\cdot
3^{3}}\frac{q^{3}h}{\tilde{f}^{2}}\,,\end{split}$ (5)

and the mass spectrum of the form

$\begin{split}m^{2}{{}_{i}}=\alpha_{i}\frac{hq^{3}}{\tilde{f}^{2}}\end{split}$
(6)

for $i=s,\tau$ and for $\alpha_{i}$ a numerical value greater than 1. By
restricting to $\tau=-\frac{6}{5}\frac{\tilde{f}}{q}$, the zeros of the
axiodilaton $s$ are elements of the splitting field of the polynomial

$\begin{split}p(s)=(hs+\tilde{f})(2hs+\tilde{f})(8hs-\tilde{f})=0,\end{split}$
(7)

for $f,h,q\in\mathbb{Z}$ as expected from Dirac quantization. Thus, the
polynomial already splits in the field $\mathbb{Q}$ and the set of
automorphisms only contains the identity element. Then the Galois group is
trivial:

$\begin{split}\text{Gal}\,p(s)=e.\end{split}$ (8)

This implies that all vacua are isolated to each other by any automorphism and
it is not possible to connect a vacuum to any other from an algebraic point of
view. In Figure 1 the dashed circles correspond to the automorphisms of the
vacua with a tachyonic direction, whereas the solid one corresponds to the
non-tachyon case.

Notice also that the physical vacuum requires that

$\begin{split}\tilde{f}<0\,\quad\text{and}\quad h,q>0\,,\end{split}$ (9)

with a $D3$-brane tadpole contribution given by

$\begin{split}N_{D3}^{\text{flux}}=\tilde{f}h.\end{split}$ (10)

Thus, since the automorphisms are trivial, the tadpole remains unchanged,
implying that $N_{D3}^{\text{flux}}<0$ and in turn requires the presence of D3
branes (or positively charged orientifold planes).

Re $s$Im
$s$$e$$\frac{\tilde{f}}{8h}$$e$$-\frac{\tilde{f}}{2h}$$e$$-\frac{\tilde{f}}{h}$
Figure 1: Roots of p(s) and the trivial automorphisms. Dashed lines represent
the automorphism on non-physical roots, while the continuous one is the
automorphism to the unique physical root.

Notice that we have selected the construction of the Galois group for the
polynomial generated by the roots of $s$ while evaluating at some fixed value
of $\tau$. Although we can perform the same procedure for $\tau$ and in this
case obtain that $\text{Gal}~{}p(\tau)=e$, in the subsequent examples we shall
restrict our study to the polynomial generated by the critical values on the
$s$-direction.

###  3.2 Uplift to dS

Let us now consider the case where an $\overline{D3}$ is added to the scalar
potential to uplift the free-tachyon vacua into a Minkowski or dS one. The
anti $D3$-brane contribution to the scalar potential is given by

$\begin{split}V_{D3}=\frac{A_{\overline{D3}}}{\mathcal{V}^{4/3}}\,,\end{split}$
(11)

with ${\cal V}=(2\tau)^{3/2}$. To find the critical points, we first consider
the set of algebraic equations

$\begin{split}\partial_{s}V=0\,,\partial_{\tau}V=0\,,V=\Lambda,\end{split}$
(12)

where the condition $V=\Lambda$ can be solved for $A_{\overline{D3}}$ being a
function of $\tau$ and $s$. Thus in the conditions of
$\partial_{s}V=\partial_{\tau}V=0$ the dependence of $A_{\overline{D3}}$ is
removed resulting in two polynomials in $s$ and $\tau$. The condition
$\partial_{s}V=0$ can be solved for $\tau$ since is a polynomial of degree 2,
and these roots are employed to leave $\partial_{\tau}V=0$ as a polynomial
depending only on $s$. The roots of $s$ now are contained in the splitting
field defined by the reducible polynomial over $\mathbb{Q}$,

$\displaystyle f(s)=25\tilde{f}^{2}q_{1}^{6}-576\tilde{f}^{3}\Lambda
q^{3}s+\left(-864\tilde{f}^{4}\Lambda^{2}-180\tilde{f}^{2}h\Lambda
q^{3}\right)s^{2}+$ (13)
$\displaystyle\left(-1728\tilde{f}^{3}h\Lambda^{2}+180\tilde{f}h^{2}\Lambda
q_{1}^{3}\right)s^{3}+1728\tilde{f}h^{3}\Lambda^{2}s^{5}+864h^{4}\Lambda^{2}s^{6}=0.$

This non-monic polynomial with integer coefficients can be transformed into a
monic polynomial $p(\tilde{s})$ with discriminant (see Appendix for details
and definitions)

$\displaystyle\text{disc}\,p(\tilde{s})$
$\displaystyle=-a\tilde{f}^{10}q^{12}h^{94}\Lambda^{50}\sum_{i=0}^{7}a_{i}\tilde{f}^{2i}h^{6-i}\Lambda^{i}q^{18-3i}$
(14)

with

$\begin{split}a&=2^{122}\cdot 2^{78}\\\ a_{0}&=3^{3}\cdot 5^{13}\\\
a_{1}&=2^{4}\cdot 3^{2}\cdot 5^{9}\cdot 11\cdot 773\\\ a_{2}&=2^{7}\cdot
5^{5}\cdot 114296701\\\ a_{3}&=2^{10}\cdot 3^{3}\cdot 5^{4}\cdot 7976063\\\
a_{4}&=2^{13}\cdot 3^{9}\cdot 5^{2}\cdot 1770757\\\ a_{5}&=2^{20}\cdot
3^{9}\cdot 11^{1}\cdot 35869\\\ a_{6}&=2^{25}\cdot 3^{12}\cdot 11^{2}\\\
\end{split}$ (15)

Clearly, $\text{disc}\,p(\tilde{s})$ is not a perfect square. This rules out
the alternating group $\text{A}_{6}$, as well as its subgroups, to be the
Galois group associated to $p(s)$.

Besides, for $\Lambda=0$ corresponding to a Minkowski vacuum, the discriminant
becomes zero, indicating that there is a degeneracy on different roots of
$p(\tilde{s})$. This degeneracy is broken once $\Lambda$ takes non-zero
values.

Therefore, for $\Lambda\neq 0$, to obtain the Galois group of $p(\tilde{s})$
we check the chain

$\begin{split}\left[(S_{3})^{2}\right]^{2}2\rightarrow
2^{3}\left[S_{3}\right]\rightarrow\sqrt{\Delta
p(\tilde{s})}\in\mathbb{Z}\rightarrow PGL(2,5)\rightarrow{\bf
S_{6}}\,,\end{split}$ (16)

which has a trimmed invariant polynomial over $S_{6}$ given by

$\begin{split}\left(\tilde{s}_{1}\tilde{s}_{2}+\tilde{s}_{3}\tilde{s}_{5}+\tilde{s}_{4}\tilde{s}_{6}\right)\left(\tilde{s}_{1}\tilde{s}_{3}+\tilde{s}_{4}\tilde{s}_{5}+\tilde{s}_{2}\tilde{s}_{6}\right)\cdot\\\
\left(\tilde{s}_{3}\tilde{s}_{4}+\tilde{s}_{1}\tilde{s}_{6}+\tilde{s}_{2}\tilde{s}_{5}\right)\left(\tilde{s}_{1}\tilde{s}_{5}+\tilde{s}_{2}\tilde{s}_{4}+\tilde{s}_{3}\tilde{s}_{6}\right)\left(\tilde{s}_{1}\tilde{s}_{4}+\tilde{s}_{2}\tilde{s}_{3}+\tilde{s}_{5}\tilde{s}_{6}\right)=0\,,\end{split}$
(17)

which representative cosets listed in stauduhar1973determination . Altogether
leads us to the conclusion that the Galois group for $p(s)$ with $\Lambda\neq
0$ is

$\begin{split}\text{Gal}\,p(s)=\mathbf{S_{6}},\end{split}$ (18)

which is non-solvable and transitive. By applying the Complex Conjugate Root
Theorem, we can deduce that the polynomial $p(s)$ includes three pairs of
complex conjugate roots. Consequently, the physical vacuum state is contained
in at least two real roots, which can be connected by automorphisms in the
case that is absolute value is equal for the two roots. However, numerical
inspection locates the two real positive roots in different absolute values,
indicating that the real root remains isolated from any automorphism of the
Gal $(p(s))$.

We also notice by a numerical inspection, that the mass of the moduli requires
that the sign of the RR flux $\tilde{f}$ is opposite to the RR flux in an AdS
vacuum. Namely, the free tachyon vacuum requires that

$\begin{split}\tilde{f}>0,\quad\text{and}\quad h,q>0\,,\end{split}$ (19)

which implies that the tadpole cancellation conditions transform into,

$\begin{split}N_{D3}^{\text{flux}}=\tilde{f}h\,.\end{split}$ (20)

Thus, the tadpole is canceled by adding negatively charged objects such as
orientifold planes. In this case, the non-triviality of the Galois group could
imply the existence of automorphisms connecting different branches of the
scalar potential. However, there are two physical vacua that are kept isolated
from the other roots. The different conditions on the number of roots, the
automorphisms, and the corresponding Galois Groups are shown in Figure 2.

AdS, $\Lambda<0$3 Roots, Gal$(p)=e$, SM, $\Lambda=0$3 RootsdS, $\Lambda>0$6
Roots, Gal$(p)=S_{6}$, NS$\overline{D3}$PhysicalAutomorphismPhysical Figure 2:
There are 3 different stable AdS vacua but only one is physical. The Galois
group is the identity $e$. After uplifting to dS by the addition of anti
$D3$-branes, the number of roots increases to 6. Four of them are connected
through automorphisms of $S_{6}$ and there are two roots isolated. S=Solvable,
NS=Non-solvable.

##  4 Example 2: A model with one complex structure

Let us consider a model in which the uplifting mechanism operates within a
complex structure, incorporating non-geometric fluxes as the one studied in
Blumenhagen:2015kja ; Blumenhagen:2015xpa . In this scenario, the
superpotential takes the following form

$\begin{split}W=-ifU+iSh_{0}-3ihSU^{2}-iqT,\end{split}$ (21)

with Kähler moduli

$\begin{split}K=-3\log\left(T+\bar{T}\right)-\log\left(S+\bar{S}\right)-3\log\left(U+\bar{U}\right).\end{split}$
(22)

In this case, the scalar potential takes the simple form where the vev’s of
the axionic partners are set to zero ($\rho=v=c=0$)

$\begin{split}V=\frac{1}{96s\tau^{3}u^{3}}\left(q^{2}\tau^{2}-18hqs\tau
u^{2}+3h_{0}^{2}s^{2}+9h^{2}s^{2}u^{4}-6fhsu^{3}+f^{2}u^{2}\right)\,.\end{split}$
(23)

To obtain the univariate polynomials where the vevs are contained, we consider
the derivative of the scalar potential with respect to each one of the real
moduli, where we obtained a set of three polynomial equations that can be
solved.

###  4.1 AdS free tachyon

The tachyon-free AdS physical vevs (the vevs that are real and positive) are
given by

$\begin{split}s_{0}=-\frac{f}{4}\sqrt{\frac{5}{h_{0}h}},\quad\tau_{0}=-\frac{f}{2q}\sqrt{\frac{5h_{0}}{h}},\quad
u_{0}=\frac{1}{3}\sqrt{\frac{5h_{0}}{h}}\end{split}$ (24)

with scalar potential at the minima given by

$\begin{split}V_{\text{AdS}}=-\frac{3^{2}}{10^{2}}\frac{q^{3}h^{2}}{f^{2}h}\sqrt{\frac{h}{5h_{0}}}\,,\end{split}$
(25)

and the mass spectrum

$\begin{split}m^{2}_{i}=\alpha_{i}\frac{q^{3}h^{2}}{f^{2}h_{0}}\sqrt{\frac{h}{5h_{0}}}\end{split}$
(26)

with $i=s,u,\tau$ and where $\alpha_{i}$ is a positive rational number. Now we
proceed to construct a degree 12 polynomial $p$ on $s$ using the Gröbner basis
( i.e., all the roots obtained by replacing for instance $u$ and $\tau$, are
contained in $p(s)$) and we observe that the SUSY minimum vevs of the saxionic
partners are elements of the field extensions

$\begin{split}\tau_{0},u_{0},s_{0}\in\mathbb{Q}\left(\sqrt{5hh_{0}},i\right)\end{split}$
(27)

for $\sqrt{5h_{0}h}\notin\mathbb{Q}$, which is always true since the Dirac
quantization condition. This is, since $h,h_{0}\in 2\mathbb{Z}$ (we consider
only even fluxes since later on we shall require the presence of orientifold
planes), thus its product has to be a multiple of 2. Thus, the product of an
odd number times an even number is never the square of a number, and in
consequence, $\sqrt{5hh_{0}}$ does not belong to $\mathbb{Q}$.

Now, let us determine the Galois group of the complete set of critical points,
which are contained in $p(s)$. This degree 12 polynomial is reducible by
Eisenstein criterium into two polynomials of degree 2 and one polynomial of
degree 8. The degree two polynomials

$\begin{split}g_{1}(s)=16hh_{0}s^{2}+5f,\quad
g_{2}(s)=16hh_{0}s^{2}-5f,\end{split}$ (28)

which can be transformed into a monic polynomial with integer coefficients

$\begin{split}\tilde{g}_{1}(s)=\tilde{s}^{2}+80f^{2}hh_{0}\quad\tilde{g}_{2}(s)=\tilde{s}^{2}-80f^{2}hh_{0}\end{split}$
(29)

have $\text{disc}\,g_{1,2}=\pm 320f^{2}hh_{0}$, which is not a perfect square,
and thus defining the Galois group of them to be $\mathbb{Z}_{2}$. The degree
8 polynomial

$\begin{split}g_{3}(s)=10616832h^{4}h_{0}^{4}s^{8}-20295f^{4}h^{2}h_{0}^{2}s^{4}+125f^{8}\end{split}$
(30)

can be transformed into the monic polynomial

$\begin{split}\tilde{g}_{3}=\tilde{s}^{8}+2^{51}\cdot 3^{1}4\cdot 5\cdot
11\cdot 41\cdot(f^{4}h^{14}h_{0}^{14})\tilde{s}^{4}+2^{119}\cdot 3^{28}\cdot
5^{3}\cdot(f^{8}h^{28}h^{28})\end{split}$ (31)

with discriminant $\text{disc}\,g_{3}=2^{781}\cdot 3^{208}\cdot 5^{17}\cdot
13^{4}\cdot 83^{8}\cdot f^{56}\cdot h^{196}\cdot h_{0}^{196}$ which is not a
perfect square, and thus the group $A_{8}$ and its subsets are excluded. In
order to determine the corresponding Galois group, one can test a chain of
subgroups of $S_{8}$ applying the criteria given in references
wildsmithcontribution . Following the notation for transitive groups given in
transitivegroups  we find that the relevant chain of subgroups to be analyzed
is given by

$\begin{split}\left[S_{4}^{2}\right]2\rightarrow\left[2^{4}\right]S_{4}\rightarrow\frac{1}{2}2^{4}S_{4}\rightarrow{\bf\frac{1}{2}\left[2^{4}\right]eD_{4}}\rightarrow\frac{1}{2}\left[2^{4}\right]4\,,\end{split}$
(32)

which has trimmed invariant polynomial over $\frac{1}{2}2^{4}S(4)$ given
by222See references wildsmithcontribution , page transitivegroups  for
notation and the application of these criteria.

$\begin{split}F_{D4\rtimes
D_{4}}=\tilde{s}_{1}\tilde{s}_{2}\tilde{s}_{5}+\tilde{s}_{1}\tilde{s}_{6}\tilde{s}_{5}+\tilde{s}_{4}\tilde{s}_{8}\tilde{s}_{5}+\tilde{s}_{2}\tilde{s}_{3}\tilde{s}_{6}+\tilde{s}_{3}\tilde{s}_{4}\tilde{s}_{7}+\tilde{s}_{2}\tilde{s}_{6}\tilde{s}_{7}+\tilde{s}_{1}\tilde{s}_{4}\tilde{s}_{8}+\tilde{s}_{3}\tilde{s}_{7}\tilde{s}_{8}\end{split}$
(33)

with coset representatives of $\frac{1}{2}\left[2^{4}\right]4$ with respect to
$\frac{1}{2}\left[2^{4}\right]eD_{4}$ given by $\\{e,(2,4,6,8)\\}$, yielding

$\begin{split}F&=-2^{43}\cdot 3^{10}\cdot
5(1+i)\sqrt[4]{2}\sqrt{3}f^{3}h^{21/2}h_{0}^{21/2}\left(\sqrt[4]{\chi}+\sqrt[4]{\bar{\chi}}\right)\\\
(2,4,6,8)F&=2^{43}\cdot 3^{10}\cdot
5(1+i)\sqrt[4]{2}\sqrt{3}f^{3}h^{21/2}h_{0}^{21/2}\left(\sqrt[4]{\chi}+\sqrt[4]{\bar{\chi}}\right)\end{split}$
(34)

where $\chi=451+249\sqrt{39}i$. Thus, since the $F$ invariant does not contain
an integer we conclude that the final Galois group of
$p(s)=g_{1}(s)g_{2}(s)g_{3}(s)$ is a subset of

$\begin{split}\text{Gal}\,(p(s))\subset\frac{1}{2}\left[2^{4}\right]4\otimes\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}\end{split}$
(35)

which is solvable and non-transitive. The roots of the defining polynomial are
shown in Figure 3.

Re $s$Im
$s$$\sigma$$\frac{f}{4}\sqrt{\frac{5}{h_{0}h}}$$-\frac{f}{4}\sqrt{\frac{5}{h_{0}h}}$$\sigma$$i\frac{f}{4}\sqrt{\frac{5}{h_{0}h}}$$-i\frac{f}{4}\sqrt{\frac{5}{h_{0}h}}$
Figure 3: 12 Roots of p(s) for the AdS case and the trivial automorphisms. The
blue lines are the automorphisms connecting the physical vacua contained in
the positive part of the x-axis. The red dashed lines are the automorphism
connecting the two non-physical vacua, as well as the eight red points that
contain the roots of the eight-degree polynomial.

The physical vev’s are contained in the field extension related to the
$\mathbb{Z}_{2}$ subgroup of the Galois group. Thus, the fixed subgroups are
composed of two elements, i.e., $\\{e,r\\}$ on the physical branch. The non-
physical vev’s are contained in the field extension dual to the polynomial of
degree 8 and contain a branch with two Kummer extensions of polynomials of
degree four over the field extension $\mathbb{Q}(\sqrt{39})$. To define a
simple extension field over $\mathbb{Q}(\sqrt{39})$, lets consider

$\begin{split}\xi=\dfrac{5}{2^{18}3^{2}}\dfrac{f^{4}}{(hh_{0})^{2}}\chi\end{split}$
(36)

which belongs to $\mathbb{Q}(\sqrt{39})$. Thus, only one extension field
contains a vev that is real and positive for the moduli.

Now, let us focus on the subfields containing the physical vevs. In this case,
the field extensions depend only on the NS fluxes since the vev depends
linearly on the non-geometric fluxes as well as the RR fluxes. This is, a
general element of the field extension
$\mathbb{Q}\left(\sqrt{5hh_{0}},\sqrt{-5hh_{0}}\right)$ can be written as,

$\begin{split}\alpha=a_{0}+\sqrt{5hh_{0}}a_{1}+ia_{2}+i\sqrt{5hh_{0}}a_{3}\end{split}$
(37)

thus $\left[\mathbb{Q}\left(\sqrt{5hh_{0}},i\right):\mathbb{Q}\right]=4$ thus
by the Lagrange theorem the subgroups related to the subfields have a
dimension of 2 and the non-geometric as well as the RR flux dependence is
contained in the coefficients $a_{i}$. For instance the choice
$a_{0}=a_{2}=a_{3}=0$ and $a_{1}=\frac{f_{1}}{4hh_{0}}$ gives the a vev of
$s_{0}$. Besides, from the point of view of the Galois theory, the
automorphism $\sigma_{1}(\sqrt{5hh_{0}})=-\sqrt{5hh_{0}}$.

In this case, the physical vacua require that

$\begin{split}f<0,\quad\text{and}\quad h_{0},\,h,\,q\,>0\end{split}$ (38)

which implies that the tadpole contribution is given by

$\begin{split}N_{D3}=fh\end{split}$ (39)

from which one notice that D3-branes are required. As discussed previously,
the branch that contained the physical vacua connects two distinct vacua via a
reflection automorphism. This implies a corresponding change in the
orientation of the cycles over which the corresponding RR fluxes are supported
on.

###  4.2 Uplift to Minkowski

Now, to uplift the AdS vacua to Minkowski vacua, we again consider the
contribution of a $\overline{\text{D3}}$ brane given by $V_{\overline{D3}}$
with the condition of having $V=0$. The critical points are then given at

$\begin{split}s_{0}=\frac{1}{3^{1/4}}\frac{f}{\left(3hh_{0}\right)^{1/2}},\quad\tau_{0}=\frac{f}{3^{1/4}q}\left(\frac{h_{0}}{h}\right)^{1/2},\quad
u_{0}=\frac{1}{3^{1/4}}\left(\frac{h_{0}}{h}\right)^{1/2}\end{split}$ (40)

which belongs to the extension
$\mathbb{Q}\left(\sqrt{3hh_{0}},3^{1/4}\right)$. The minimal polynomial
containing all the roots is given by

$\begin{split}p(s)=s^{4}-\frac{f^{4}}{27h_{0}^{2}h^{2}}=0\,,\end{split}$ (41)

which clearly is a non-monical polynomial. Lets consider the monic polynomial
$p(\tilde{s})=(27h_{0}^{2}h^{2})^{4}p\left(\frac{s}{27h_{0}^{2}h^{2}}\right)$,
defined as

$\begin{split}\tilde{s}^{4}-19683f^{4}h^{6}h_{0}^{6}=0\end{split}$ (42)

where $\tilde{s}=\frac{s}{27h_{0}^{2}h^{2}}$ and the roots of the modified
polynomial are contained in the same splitting field as $p(s)$.

Thus, the roots are contained in a degree 4 polynomial, with integer
coefficients, which is irreducible over $\mathbb{Q}$ by the Eisenstein
criterium. In consequence, the Galois group must be a transitive subgroup of
$S_{4}$. There are five transitive subgroups of $S_{4}$, and since the
discriminant of the defining polynomial is not a square, the Galois group
shall be either $D_{4}$ or $\mathbb{Z}_{4}$. Thus, lets take test the $D_{4}$
invariant polynomial relative to $S_{4}$

$\begin{split}F_{D_{4}}=\tilde{s}_{1}\tilde{s}_{3}+\tilde{s}_{2}\tilde{s}_{4}\,,\end{split}$
(43)

for $\tilde{s}_{i}$ any root of the defining polynomial, with coset
representative of $D_{4}$ with respect to $S_{4}$ as $\\{(e),(23),(34)\\}$,
yielding

$\begin{split}(\text{e})F&=\tilde{s}_{1}\tilde{s}_{3}+\tilde{s}_{2}\tilde{s}_{4}=-162i\sqrt{3}f^{2}h^{3}h_{0}^{3}\\\
(23)F&=\tilde{s}_{1}\tilde{s}_{2}+\tilde{s}_{3}\tilde{s}_{4}=162i\sqrt{3}f^{2}h^{3}h_{0}^{3}\\\
(34)F&=\tilde{s}_{1}\tilde{s}_{4}+\tilde{s}_{2}\tilde{s}_{3}=0\\\ \end{split}$
(44)

since there exists one rational conjugate evaluation of $F_{D_{4}}$ then we
conclude that the Galois group for the Minkowski vacua, is given by

$\begin{split}\text{Gal}\,(f(s))=\mathbf{D_{4}}\,,\end{split}$ (45)

which is transitive and solvable333 See Appendix, section 2 for more details
about the decision tree related to $S_{4}$..

In this case, the tachyon-free physical vacua require that

$\begin{split}f>0,\quad\text{and}\quad h,\,h_{0},\,q\,>0\,,\end{split}$ (46)

with a tadpole contribution given by

$\begin{split}N_{D3}^{\text{flux}}=fh.\end{split}$ (47)

Thus, the tadpole becomes positive, requiring O3 planes to cancel it. In this
case, the subgroup lattice diagram is given by Figure 4. In this case, all the
vacua are connected by automorphisms of the D4 group.

$e$$Z_{2}$$Z_{2}$$Z_{2}$$Z_{2}^{2}$$Z_{4}$$Z_{2}^{2}$$D_{4}$22 Figure 4:
Subgroup lattice diagram of D4. The small gray index indicates that there are
two copies of the group

###  4.3 Uplift to dS

Now, let us explore the Galois group for the critical points obtained by
solving the set of algebraic equations,

$\begin{split}\partial_{u}V=\partial_{\tau}V=\partial_{s}V=0\,,\quad
V=\Lambda\end{split}$ (48)

where $\Lambda>0$ is the cosmological constant and the contribution to the
scalar potential coming from $\overline{D3}$ given by $V_{\overline{D3}}$. In
this case, the numerator $A_{\overline{D3}}$ which is related to the warping
factor of the metric can be considered as a parameter that is adjusted to
achieve the positive value of the vacuum and take the value

$\begin{split}A_{\overline{D3}}=\tau^{2}\left(\Lambda-V\right)\end{split}$
(49)

thus, the derivatives of the scalar potential can be written only in terms of
the moduli. In order to eliminate the variables $\tau$ and $u$ we calculate
the Gröbner basis for the ideal generated by the derivatives of the scalar
potential giving a polynomial of degree 34 for the axiodilaton, given by

$\begin{split}p(s)&=-f^{22}\Lambda^{2}q^{18}-512f^{19}h_{0}^{4}\Lambda^{3}q^{15}s^{5}-5109f^{18}h^{2}h_{0}^{2}\Lambda^{2}q^{18}s^{4}-32f^{17}h^{4}\Lambda
q^{21}s^{3}\\\
&-35831808f^{16}h_{0}^{8}\Lambda^{4}q^{12}s^{10}+8272512f^{15}h^{2}h_{0}^{6}\Lambda^{3}q^{15}s^{9}-5748880f^{14}h^{4}h_{0}^{4}\Lambda^{2}q^{18}s^{8}\\\
&-96672f^{13}h^{6}h_{0}^{2}\Lambda
q^{21}s^{7}-18345885696f^{13}h_{0}^{12}\Lambda^{5}q^{9}s^{15}-2304f^{12}h^{8}q^{24}s^{6}\\\
&+174439968768f^{12}h^{2}h_{0}^{10}\Lambda^{4}q^{12}s^{14}+17294212800f^{11}h^{4}h_{0}^{8}\Lambda^{3}q^{15}s^{13}\\\
&-251138880f^{10}h^{6}h_{0}^{6}\Lambda^{2}q^{18}s^{12}-320979616137216f^{10}h_{0}^{16}\Lambda^{6}q^{6}s^{20}-6656256f^{9}h^{8}h_{0}^{4}\Lambda
q^{21}s^{11}\\\
&-50008591171584f^{9}h^{2}h_{0}^{14}\Lambda^{5}q^{9}s^{19}+62208f^{8}h^{10}h_{0}^{2}q^{24}s^{10}-37252825841664f^{8}h^{4}h_{0}^{12}\Lambda^{4}q^{12}s^{18}\\\
&-2853603758592f^{7}h^{6}h_{0}^{10}\Lambda^{3}q^{15}s^{17}-164341563462254592f^{7}h_{0}^{20}\Lambda^{7}q^{3}s^{25}\\\
&-23145108480f^{6}h^{8}h_{0}^{8}\Lambda^{2}q^{18}s^{16}-19195373587267584f^{6}h^{2}h_{0}^{18}\Lambda^{6}q^{6}s^{24}\\\
&-3418860141674496f^{5}h^{4}h_{0}^{16}\Lambda^{5}q^{9}s^{23}+2140954131336192f^{4}h^{6}h_{0}^{14}\Lambda^{4}q^{12}s^{22}\\\
&+16269814782763204608f^{3}h^{2}h_{0}^{22}\Lambda^{7}q^{3}s^{29}+2211549555185418240f^{2}h^{4}h_{0}^{20}\Lambda^{6}q^{6}s^{28}\\\
&+567964443325551869952h^{2}h_{0}^{26}\Lambda^{8}s^{34}.\end{split}$ (50)

This polynomial is normalized to obtain a monic polynomial with integer
coefficients, which is the polynomial that contains the roots of the
axiodilaton at the vacuum. This polynomial is too big to be examined by
trimmed invariants. However, for specific values of the fluxes, it is possible
to determine the Galois group employing the Magma package. In this case, the
Galois group is the symmetric group acting on a set of cardinality 34, this is

$\begin{split}\text{Gal}\left(p(s)\right)=S_{34}\end{split}$ (51)

which is non-solvable and transitive. As in the Minkowski case, the physical
vacua requires the choice of orientation for the fluxes

$\begin{split}f>0,\quad\text{and}\quad h,\,h_{0},\,q\,>0\,,\end{split}$ (52)

with a tadpole contribution given by

$\begin{split}N_{D3}^{\text{flux}}=fh,\end{split}$ (53)

requiring O3 planes to cancel it. We show the main characteristics of the
uplifting process in Figure 5.

AdS, $\Lambda<0$12 Roots,
Gal$~{}p(s)=G\otimes\mathbb{Z}_{2}\otimes\mathbb{Z}_{2}$, SM, $\Lambda=0$4
Roots, Gal$~{}p(s)=D_{4}$, SdS, $\Lambda>0$34 Roots, Gal$~{}p(s)=S_{34}$,
NS$\overline{D3}$Physical… Figure 5: dS vacua are related to 34 roots of a
polynomial with a Galois group given by $S_{34}$ which is a non-solvable
group, contrary to the case of AdS and Minkowski. Through the lifting process,
one can notice that some roots could be related and degenerated, although a
much more detailed analysis is required to clearly elucidate those aspects.
$G=\frac{1}{2}\left[2^{4}\right]4$, S= Solvable, NS= Non-Solvable.

##  5 Example 3: A case with non-BPS states

Let us consider the scalar potential generated in the presence of non-BPS
states $\widehat{D5}$ and $F_{5}$ as considered in Damian:2022xfk

$\begin{split}\tilde{V}=\frac{A_{\widehat{D5}}}{\sqrt{s}\tau^{5/2}}+V\end{split}$
(54)

where

$\begin{split}V=\frac{A_{F_{3}}}{s\tau^{3}}+\frac{A_{F_{5}}}{\tau^{4}}+\frac{A_{H_{3}}s}{\tau^{3}}+\frac{A_{O3}}{\tau^{3}}.\end{split}$
(55)

The scalar potential is split into the contribution coming from the standard
RR and NS fluxes, a F5 flux forms coming from the torsional NS and RR fluxes
warped on torsional cycles, and the contribution from orientifold 3-planes.

###  5.1 AdS free tachyon

In the absence of the non-BPS states A${}_{\widehat{D5}}$, the scalar
potential admits a stable AdS vacua obtained from the roots of the algebraic
equations

$\begin{split}\partial_{s}V=\partial_{\tau}V=0\,,\end{split}$ (56)

the vev’s of the saxionic fields are thus

$\begin{split}(s_{0},\tau)=\left[\left(-\sqrt{\frac{A_{F_{3}}}{A_{H_{3}}}},\frac{4}{3}\frac{A_{F_{5}}}{\sqrt{4A_{F_{3}}A_{H_{3}}}-A_{O3}}\right),\left(\sqrt{\frac{A_{F_{3}}}{A_{H_{3}}}},-\frac{4}{3}\frac{A_{F_{5}}}{\sqrt{4A_{F_{3}}A_{H_{3}}}+A_{O3}}\right)\right]\end{split}$
(57)

where the physical vacua are related to the second vev. The vacua is AdS with
value

$\begin{split}V_{\text{min}}=-\frac{1}{3}\frac{A_{F_{5}}}{\tau_{0}^{4}}\,,\end{split}$
(58)

and mass spectrum

$\begin{split}m^{2}=\left(\frac{2A_{H_{3}}^{1/2}}{s_{0}\tau_{0}},\frac{4A_{F_{5}}}{\tau_{0}^{6}}\right)\,.\end{split}$
(59)

The physical vacua requires that $A_{F_{3}},A_{H_{3}},A_{F_{5}}>0$, and for
the flux contribution being a rational number it is possible to conclude that
the vevs are contained in the field extension

$\begin{split}s_{0},\tau_{0}\in\mathbb{Q}\left(\sqrt{A_{F_{3}}A_{H_{3}}}\right)\,,\end{split}$
(60)

and the axiodilaton vev’s are elements of the roots of a degree 2 polynomial

$p(s)=s^{2}-\frac{A_{F_{3}}}{A_{H_{3}}}=0\,,$ (61)

this polynomial can be normalized to be a monic polynomial with integer
coefficients of the form

$\begin{split}p(\tilde{s})=\tilde{s}^{2}-A_{F_{3}}A_{H_{3}}=0\,,\end{split}$
(62)

with discriminant, $\Delta=4A_{F_{3}}A_{H_{3}}$ which could be a perfect
square if $A_{F_{3}}=A_{H_{3}}$. However, this case corresponds to a vev for
the axionic field equal to 1 which could lead to an inconsistency of the
tadpole cancellation conditions. By Eisenstein criterium, the defining
polynomial for $\tilde{s}$ is irreducible over $\mathbb{Q}$ and hence its
Galois group is a transitive group, subset of the symmetric group $S_{2}$. The
cyclic group $\mathbb{Z}_{2}$ is the only transitive subgroup. The splitting
field is $\mathbb{Q}\left(\sqrt{\frac{A_{F_{3}}}{A_{H_{3}}}}\right)$. Thus the
Galois group containing the vev of the AdS vacuum is

$\begin{split}\text{Gal}~{}p(s)=\mathbb{Z}_{2}\,,\end{split}$ (63)

which is transitive and solvable. The roots of the physical vacua are shown in
Figure 6.

Re $s$Im
$s$$\sigma$$-\sqrt{\frac{A_{F_{3}}}{A_{H_{3}}}}$$\sqrt{\frac{A_{F_{3}}}{A_{H_{3}}}}$
Figure 6: Roots of p(s) and the trivial automorphisms.

In this simple case the group $\mathbb{Z}_{2}$ contains two automorphisms
related to the identity element and a reflection, this implies that the two
vacua contained in the splitting field of $p(s)$ are connected by an
automorphism $\sigma(s_{1})=-s_{2}$ where $s_{1}$ are the two roots of the
$p(s)$. However, by the action of this automorphism, the vev of the
axiodilaton changes of sign making it a non-physical vacuum.

###  5.2 Uplift to Minkowski

Let us consider the case for $A_{\widehat{D5}}\neq 0$, and its value
parametrized such that

$\begin{split}\tilde{V}=0\rightarrow
A_{\widehat{D5}}=-\sqrt{s}\tau^{5/2}V\,,\end{split}$ (64)

the vevs for the moduli s and $\tau$ are thus contained in the roots of the
algebraic equations

$\begin{split}\partial_{s}\tilde{V}=\partial_{\tau}\tilde{V}=0\,,\end{split}$
(65)

thus, eliminating $\tau$ the system is reduced to

$\begin{split}p(s)=4A_{H_{3}}s^{2}+A_{O3}s-2A_{F_{3}},\end{split}$ (66)

which is normalized to the monic polynomial with integer coefficients,

$\begin{split}p(\tilde{s})=\tilde{s}^{2}+A_{O3}\tilde{s}-8A_{F3}A_{H3}\end{split}$
(67)

with discriminant $\Delta=A_{O3}^{2}+32A_{F3}A_{H3}$. Since the polynomial
$f(\tilde{s})\in\mathbb{Q}\left[\tilde{s}\right]$ is of degree two, the Galois
group is

$\begin{split}\text{Gal}\,p(s)=\mathbb{Z}_{2}\end{split}$ (68)

which is transitive and solvable. As in the AdS case, the non-trivial
automorphism of $\mathbb{Z}_{2}$ implies the mapping of a physical to a non-
physical vacuum.

###  5.3 Uplift to dS: Scenario 1

Let us consider the uplift of the AdS vacua by considering

$\begin{split}\tilde{V}=\Lambda\rightarrow
A_{\widehat{D5}}=\sqrt{s}\tau^{5/2}\left(\Lambda-V\right)\,,\end{split}$ (69)

and the moduli vevs are contained at the zeros of the algebraic equations

$\begin{split}\partial_{s}\tilde{V}=\partial_{\tau}V=0,\end{split}$ (70)

which by eliminating $\tau$ we are left with a polynomial of degree 8 given by

$\begin{split}p(s)&=16A_{F_{3}}^{4}-80A_{F_{3}}^{3}A_{O3}s+16A_{F_{3}}^{2}\left(9A_{O3}^{2}-14A_{F_{3}}A_{H_{3}}\right)s^{2}\\\
&+12A_{F_{3}}A_{O3}\left(68A_{F_{3}}A_{H_{3}}-9A_{O3}^{2}\right)s^{3}\\\
&+(1152A_{F_{3}}^{2}A_{H_{3}}^{2}-936A_{F_{3}}A_{H_{3}}A_{O3}^{2}+256A_{F_{5}}^{3}\Lambda+27A_{O3}^{4})s^{4}\\\
&+12A_{H_{3}}A_{O3}\left(27A_{O3}^{2}-224A_{F_{3}}A_{H_{3}}\right)s^{5}+160A_{H_{3}}^{2}\left(9A_{O3}^{2}-16A_{F_{3}}A_{H_{3}}\right)s^{6}\\\
&+2816A_{H_{3}}^{3}A_{O3}s^{7}+2048A_{H_{3}}^{4}s^{8},\end{split}$ (71)

which is normalized to a monic polynomial with integer coefficients with
discriminant

$\displaystyle\Delta$
$\displaystyle=2^{542}A_{F_{3}}^{12}A_{F_{5}}^{12}A_{H_{3}}^{180}\Lambda^{4}\bigg{(}1565515579392A_{F_{3}}^{8}A_{H_{3}}^{8}-856141332480A_{F_{3}}^{7}A_{H_{3}}^{7}A_{O3}^{2}$
(72)
$\displaystyle-382205952A_{F_{3}}^{6}A_{H_{3}}^{6}\left(458752A_{F_{5}}^{3}\Lambda+23A_{O3}^{4}\right)+$
$\displaystyle
5971968A_{F_{3}}^{5}A_{H_{3}}^{5}A_{O3}^{2}\left(10367A_{O3}^{4}-69107712A_{F_{5}}^{3}\Lambda\right)$
$\displaystyle-6912A_{F_{3}}^{4}A_{H_{3}}^{4}\left(-708132732928A_{F_{5}}^{6}\Lambda^{2}+28126089216A_{F_{5}}^{3}A_{O3}^{4}\Lambda+401085A_{O3}^{8}\right)$
$\displaystyle-6912A_{F_{3}}^{3}A_{H_{3}}^{3}A_{O3}^{2}\left(-314136592384A_{F_{5}}^{6}\Lambda^{2}+5659352064A_{F_{5}}^{3}A_{O3}^{4}\Lambda+251559A_{O3}^{8}\right)$
(73)
$\displaystyle+864A_{F_{3}}^{2}A_{H_{3}}^{2}\big{(}962072674304A_{F_{5}}^{9}\Lambda^{3}+436263190528A_{F_{5}}^{6}A_{O3}^{4}\Lambda^{2}$
$\displaystyle-4393191168A_{F_{5}}^{3}A_{O3}^{8}\Lambda+47385A_{O3}^{12}\big{)}+432A_{F_{3}}A_{H_{3}}A_{O3}^{2}\big{(}-1181116006400A_{F_{5}}^{9}\Lambda^{3}$
$\displaystyle+56813027328A_{F_{5}}^{6}A_{O3}^{4}\Lambda^{2}-399810816A_{F_{5}}^{3}A_{O3}^{8}\Lambda+45927A_{O3}^{12}\big{)}$
$\displaystyle+\left(256A_{F_{5}}^{3}\Lambda+27A_{O3}^{4}\right)^{2}\left(536870912A_{F_{5}}^{6}\Lambda^{2}-3649536A_{F_{5}}^{3}A_{O3}^{4}\Lambda+729A_{O3}^{8}\right)\big{)}.$

First of all, observe that for $\Lambda=0$ the discriminant vanishes implying
the presence of degeneracy. Hence the two roots present in the Minkowski vacua
split into different roots in the dS uplifting. Second of all, since $\Delta$
is not a perfect square, the Galois group cannot be a subset of $A_{8}$, and
the decision tree for this case gives the following sequence of subgroups:

$\begin{split}\left[S_{4}^{2}\right]^{2}\rightarrow\left[2^{4}\right]S_{4}\rightarrow
PGL(2,7)\rightarrow{\bf{S_{8}}}\end{split}$ (74)

from which we conclude that the Galois group is

$\begin{split}\text{Gal}\,p(s)=S_{8}\end{split}$ (75)

which is transitive and non-solvable.

###  5.4 Uplift to dS: Scenario 2

Now, let us consider the stabilization of moduli taking $A_{\widehat{D5}}$ as
a free parameter and the moduli fixed by the roots of the algebraic equations

$\begin{split}\partial_{s}V=\partial_{\tau}V=0\,,\end{split}$ (76)

and eliminating $\tau$, the vev for the axiodilaton is fixed by the roots of

$\begin{split}p(s)&=A_{\widehat{D5}}^{2}A_{F_{5}}s^{2}-2A_{F_{3}}^{3}+12A_{F_{3}}^{2}A_{H_{3}}s^{2}+3A_{F_{3}}^{2}A_{O3}s-18A_{F_{3}}A_{H_{3}}^{2}s^{4}-6A_{F_{3}}A_{H_{3}}A_{O3}s^{3}\\\
&+8A_{H_{3}}^{3}s^{6}+3A_{H_{3}}^{2}A_{O3}s^{5},\end{split}$ (77)

which is normalized to the monic polynomial with integer coefficients

$\begin{split}p(\tilde{s})&=512A_{\widehat{D5}}^{2}A_{F_{5}}A_{H_{3}}^{9}\tilde{s}^{2}-65536A_{F_{3}}^{3}A_{H_{3}}^{15}+12288A_{F_{3}}^{2}A_{H_{3}}^{12}A_{O3}\tilde{s}+6144A_{F_{3}}^{2}A_{H_{3}}^{10}\tilde{s}^{2}\\\
&-384A_{F_{3}}A_{H_{3}}^{7}A_{O3}\tilde{s}^{3}-144A_{F_{3}}A_{H_{3}}^{5}\tilde{s}^{4}+3A_{H_{3}}^{2}A_{O3}\tilde{s}^{5}+\tilde{s}^{6},\end{split}$
(78)

with discriminant

$\displaystyle\Delta$
$\displaystyle=2^{60}A_{\widehat{D5}}^{4}A_{F_{3}}^{3}A_{F_{3}}^{2}A_{H_{3}}^{68}\bigg{(}1048576A_{\widehat{D5}}^{8}A_{F_{3}}^{4}A_{H_{3}}$
(79)
$\displaystyle+72A_{\widehat{D5}}^{6}A_{F_{3}}^{3}\left(753664A_{F_{3}}^{2}A_{H_{3}}^{2}+20224A_{F_{3}}A_{H_{3}}A_{O_{3}}^{2}-243A_{O_{3}}^{4}\right)$
$\displaystyle+27A_{\widehat{D5}}^{4}A_{F_{3}}A_{F_{3}}^{2}\left(28098560A_{F_{3}}^{3}A_{H_{3}}^{3}+5960448A_{F_{3}}^{2}A_{H_{3}}^{2}A_{O_{3}}^{2}-23328A_{F_{3}}A_{H_{3}}A_{O_{3}}^{4}-729A_{O_{3}}^{6}\right)$
$\displaystyle+15552A_{\widehat{D5}}^{2}A_{F_{3}}^{3}A_{F_{3}}A_{H_{3}}\left(94208A_{F_{3}}^{3}A_{H_{3}}^{3}+66752A_{F_{3}}^{2}A_{H_{3}}^{2}A_{O_{3}}^{2}+8060A_{F_{3}}A_{H_{3}}A_{O_{3}}^{4}-159A_{O_{3}}^{6}\right)$
$\displaystyle+186624A_{F_{3}}^{4}A_{H_{3}}\left(4A_{F_{3}}A_{H_{3}}-A_{O_{3}}^{2}\right)^{3}\left(64A_{F_{3}}A_{H_{3}}+9A_{O_{3}}^{2}\right)\bigg{)},$

which is not a perfect square. Thus, the roots are contained on a polynomial
of degree 6 which is irreducible over $\mathbb{Q}$. Hence the Galois group is
fixed to be

$\begin{split}\text{Gal}\left(p(s)\right)=S_{6}\,,\end{split}$ (80)

which is transitive and non-solvable. In Figure 7 we summarized the different
roots of the associated polynomials as well as the corresponding Galois groups
for AdS, Minkowski and dS vacua.

AdS, $\Lambda<0$2 Roots, Gal$~{}p(s)=\mathbb{Z}_{2}$M, $\Lambda=0$2 Roots,
Gal$~{}p(s)=\mathbb{Z}_{2}$dS, $\Lambda>0$6/8 Roots,
Gal$~{}p(s)=S_{6/8}$$\widehat{D5}$$\Lambda\not=0$1 physical… Figure 7: In this
case there are only two roots in AdS and Minkowski, and only one of them is
physical. As in the previous cases, once we uplift to dS, in this case by
adding non-BPS states, the degenerate roots in the Minkowski vacua split into
6 or 8 roots, fixing the Galois group to $G_{6/8}$ which is unsolvable.

##  6 Example 4: A Calabi-Yau with $h_{2,1}\neq 0$

Let’s consider the tree-level analytical models considered in
Coudarchet:2022fcl , where it is considered a non-scale type IIB
compactification on a Calabi-Yau 3-fold X3. As usual, the periods are written
as444We are using the same notation as the reference Coudarchet:2022fcl  where
the saxions are the imaginary part of the moduli, contrary to our previous
examples.

$\begin{split}\Pi^{t}=\left(\int_{B_{I}}\Omega,\int_{A^{I}}\Omega\right)\,,\end{split}$
(81)

where the cycles $\\{A^{I},B_{I}\\}$ are elements of a symplectic basis, and
the complex structure is defined as $z_{i}=X^{i}/X^{0}$ for
$i=1,\ldots,h^{2,1}$ together with the gauge $X^{0}=1$. Thus, the GVW
superpotential is expressed as

$\begin{split}W=\int_{X_{3}}\left(F_{3}-\tau
H_{3}\right)\wedge\Omega\,,\end{split}$ (82)

which is written as

$\begin{split}W=-\frac{1}{6}N_{A}^{0}\kappa_{ijk}z^{i}z^{j}z^{k}+\frac{1}{2}\kappa_{ijk}N_{A}^{i}z^{j}z^{k}+\left(N_{A}^{j}a_{ij}+N_{i}^{B}-N_{A}^{0}c_{i}\right)z^{i}-\kappa_{0}N^{0}_{A}-N_{A}^{i}c_{i}+N_{0}^{B}\,.\end{split}$
(83)

where the flux vector is defined as

$\begin{split}N=\left(\begin{matrix}f_{0}^{B}\\\ f_{i}^{B}\\\ f_{A}^{0}\\\
f_{A}^{i}\\\ \end{matrix}\right)-\tau\left(\begin{matrix}h_{0}^{B}\\\
h_{i}^{B}\\\ h_{A}^{0}\\\ h_{A}^{i}\\\
\end{matrix}\right)=\left(\begin{matrix}N_{0}^{B}\\\ N_{i}^{B}\\\ N_{A}^{0}\\\
N_{A}^{i}\\\ \end{matrix}\right)\end{split}$ (84)

which allows writing the tadpole as

$\begin{split}N_{\text{flux}}=\frac{N^{\dagger}\cdot\Sigma\cdot\Pi}{\tau-\bar{\tau}}\,.\end{split}$
(85)

On the other hand, the Kähler potential is given by

$\begin{split}K=-2\ln\mathcal{V}-\ln\left(-i(\tau-\bar{\tau})\right)-\ln\left(-i\Pi^{\dagger}\cdot\Sigma\cdot\Pi\right)\end{split}$
(86)

where $\mathcal{V}$ is the volume modulus, $\tau=b^{0}+it^{0}$ is the axio-
dilaton and the complex structures are $z^{i}=b^{i}+it^{i}$. The canonical
symplectic matrix is

$\begin{split}\Sigma=\left(\begin{matrix}0&\mathbf{1}\\\ -\mathbf{1}&0\\\
\end{matrix}\right)\end{split}$ (87)

which explicitly give us the Kähler potential

$\begin{split}K=-2\ln\mathcal{V}-\ln\left(2t^{0}\right)-\ln\left(\frac{4}{3}\kappa_{ijk}t^{i}t^{j}t^{k}-2\text{Im}\kappa_{0}\right)\,,\end{split}$
(88)

In Coudarchet:2022fcl , the authors focus on the dubbed IIB1 scenario, which
is based upon selecting the appropriate RR and NS fluxes that makes the
superpotential able to be written in the bilinear form

$\begin{split}W=\frac{1}{2}\vec{Z}^{t}M\vec{Z}+\vec{L}\cdot\vec{Z}+Q\end{split}$
(89)

which comes from an F-theory compactification in the large complex structure
region. This scenario allows writing the scalar potential into a bilinear
structure which is quite useful to express the critical point of the scalar
potential in order to systematically study the corresponding vacua. In this
scenario, the following fluxes are set to zero,

$\begin{split}f_{A}^{0}=h_{A}^{0}=h_{A}^{i}=0\quad
i\in\\{1,\ldots,h^{2,1}\\}\end{split}$ (90)

reducing the tadpole to

$\begin{split}N_{\text{flux}}=-f_{A}^{i}h^{B}_{i}\,,\end{split}$ (91)

where

$\begin{split}M=\left(\begin{matrix}0&-\vec{(}h^{B})^{T}\\\
-\vec{h}^{B}&h_{ij}\\\
\end{matrix}\,,\right)\quad\vec{L}=\left(-h_{0}^{B},f_{i}^{B}+a_{ij}f^{j}_{A}\right)\,,\quad
Q=f_{0}^{B}-c_{i}f^{i}_{A}\,.\end{split}$ (92)

The scalar potential constructed from Eq. 89 admits non-supersymmetric vacua
contained in the set of algebraic equations

$\begin{split}\tilde{x}^{2}&=\frac{N_{\text{flux}}}{\kappa^{H}}\hat{x}\left(\hat{x}-1\right)\,,\\\
x^{0}&=\frac{\mathcal{S}\hat{x}(\hat{x}-1)}{N_{\text{flux}}}+\tilde{x}-2\hat{x}\tilde{x}\,,\\\
\left(2\hat{x}^{3}-3\hat{x}^{2}+\alpha\right)^{2}&=16\frac{N^{3}_{\text{flux}}}{\mathcal{S}\kappa^{H}}\hat{x}^{3}\left(\hat{x}-1\right)^{3}\,.\end{split}$
(93)

where

$\begin{split}x^{0}&=\frac{4}{3}\frac{Q^{\prime}}{\text{Im}\kappa_{0}}t^{0}\,,\quad
x^{i}=\frac{4}{3}\frac{Q^{\prime}}{\text{Im}\,\kappa_{0}}t^{i}\,,\quad\alpha=\frac{2^{5}Q^{\prime
3}}{3^{2}\left(\text{Im}\,\kappa_{0}\right)^{2}\mathcal{S}}\\\
\mathcal{S}&=\kappa_{ijk}f_{A}^{i}f_{A}^{j}f_{A}^{k}\,,\quad\kappa^{H}=\kappa_{ijk}S^{il}h_{l}^{B}S^{jm}h_{m}^{B}S^{kn}h_{n}^{B}\end{split}$
(94)

and

$\begin{split}x^{i}=\hat{x}f^{i}_{A}+\tilde{x}S^{ij}h_{i}^{B}\,,\end{split}$
(95)

thus by solving Eq. 93 it is possible to obtain the vev’s of the saxionic
moduli. To look for the Galois group we transform the non-monic polynomial
into the monic polynomial with integer coefficients. As it is observed, the
roots of $\hat{x}$ are contained on a degree 6 polynomial

$\begin{split}p(\hat{x})=\kappa^{H}\mathcal{S}\left(\alpha+2\hat{x}^{3}-3\hat{x}^{2}\right)^{2}-16N_{\text{flux}}^{3}(\hat{x}-1)^{3}\hat{x}^{3}\,,\end{split}$
(96)

thus, there are 16 transitive groups of degree 6, with 5 of them being non-
solvable. However, in this case, the discriminant is

$\displaystyle\Delta$ $\displaystyle=2^{62}\cdot 3^{6}\cdot
N_{\text{flux}}^{9}\mathcal{S}^{4}(\alpha-1)^{4}\alpha^{4}(\kappa^{H})^{4}\left(4N_{\text{flux}}^{3}-\mathcal{S}\kappa^{H}\right)^{5}\left(\mathcal{S}\kappa^{H}-4N_{\text{flux}}^{3}\right)^{16}$
(97)
$\displaystyle\left(N_{\text{flux}}^{6}+4\mathcal{S}(\alpha-1)\alpha\kappa^{H}N_{\text{flux}}^{3}-\mathcal{S}^{2}(\alpha-1)\alpha(\kappa^{H})^{2}\right)$

discarding the alternating group $A_{6}$ as well as its subgroups as the
Galois group. The decision tree goes as follows

$\begin{split}\left[S_{3}^{2}\right]2\rightarrow
2^{3}\left[S_{3}\right]\rightarrow\sqrt{\nabla(f(\hat{x}))}\in\mathbb{Z}\rightarrow
PGL(2,5)\rightarrow\mathbf{S}_{6}\end{split}$ (98)

yielding the Galois group

$\begin{split}\text{Gal}\left(p(\hat{x})\right)=S_{6}\,,\end{split}$ (99)

which is non-solvable and transitive.

##  7 Conclusions

We have chosen four different scenarios in which it is possible, at least
numerically, to obtain stable de Sitter (dS) vacua by adding exotic structures
such as non-geometric fluxes or non-BPS branes, resulting in a polynomial
scalar potential. This potential is crucial to our study as we are interested
in computing the corresponding Galois group, which provides important
information about the algebraic structure of the roots of the polynomial
representing the scalar potential’s moduli. To compute the Galois group of a
specific modulus, we utilized conventional techniques reported in existing
literature, although the analysis can be extended to other moduli yielding
similar outcomes.

In all four selected examples, the presence of dS critical points is
consistently associated with the absence of analyticity caused by the
unsolvability of the Galois group, in contrast to Anti-de Sitter (AdS) and
Minkowski vacua. The solvability of the Galois group implies the possibility
of expressing the polynomial roots in terms of radicals and fractional
numbers. Notably, for all uplifted dS vacua, the Galois group becomes
unsolvable. This observation suggests a connection between the challenge of
finding analytical examples of dS vacua and the unsolvability of their
associated Galois groups.

Additionally, although the Galois group is non-solvable, we expect an
automorphism connecting different dS critical points since it relates to the
critical points associated with all vacua. However, in all cases, the physical
vacua are either isolated from the other critical points or connected to a
non-physical vacuum characterized by a negative value of the saxionic
components’ vacuum expectation value (vev). We also observe that a change in
the orientations of fluxes can be interpreted as the result of
$\mathbb{Z}_{2}$ automorphisms, which act as a bridge between the
automorphisms of Minkowski vacua. As a Minkowski vacuum continuously deforms
into a dS vacuum, its roots undergo degeneration, leading to a similar
degeneration in a branch of physical vacua and resulting in changes in vacuum
energy.

Therefore, based on our studied examples, it is expected that dS vacua in
perturbative compactifications, if they exist, are associated with a non-
solvable Galois group. If the swampland de Sitter conjecture holds true, one
could conclude that all vacua in the landscape are linked to a solvable Galois
group, at least within a perturbative framework. A more detailed study is
undoubtedly required to confirm this association.

##  8 Appendix

###  8.1 A (slightly more formal) brief introduction to Galois theory

For those who want to read a little more formal description of the steps to
compute the Galois group, in this section, we provide more specific details. A
reader not interested in these details can safely skip this appendix. For a
more detailed description, the interested reader must check
brzezinski2018galois ; stewart2022galois . Overall, computing the Galois group
of a given polynomial can be a complex and involved process that requires
knowledge of algebraic structures, group theory, and field theory.

The Galois theory gives a relation between the roots of a polynomial $p(x)$
and the group action of the splitting fields related to such roots. The
splitting field of a polynomial $p(x)$, denoted $L[p]$ is defined as the
smallest field $\mathbb{Q}$ generated by the roots $\alpha_{i}$ of $p(s)$ and
is denoted as,

$L[p]=\mathbb{Q}\left(\alpha_{1},\alpha_{2},\ldots\right),$ (100)

The relation between $L\left[p\right]$ and $\\{\alpha_{i}\\}$ is codified in
the so-called group of automorphisms $\text{Aut}[L]$ of $L[p]$, which is
constructed by bijective function $\sigma:L\rightarrow L$ such that, for all
$\alpha,\beta\in L$,

  * •

$\sigma(\alpha+\beta)=\sigma(\alpha)+\sigma(\beta)$,

  * •

$\sigma(\alpha\beta)=\sigma(\alpha)\sigma(\beta)$,

  * •

$\sigma(\alpha)=\alpha\quad\forall\alpha\in K\subset L$,

where the third property defines the so-called K-automorphism. Thus, the group
of all K-automorphisms of L is denoted as $\text{Gal}(L/K)$ and is known as
the Galois group of L over K. Notice that if $\alpha_{i}$ is a root of $p$,
thus

$\begin{split}p(\sigma(\alpha_{i}))=\sigma(p(\alpha_{i}))=0\,.\end{split}$
(101)

and similarly $\sigma^{-1}\left(\alpha_{i}\right)$ is also a root of $p$,
allowing us to abuse the notation and refer to Gal$(L/K)$ as the Galois group
of the polynomial $p(x)$ denoted by Gal $~{}p(x)$.

In order to compute the Galois group of the splitting field $L\left[p\right]$,
it is required for the latter to be normal and separable. If an irreducible
polynomial contains a root in $L$, it must contain all the roots. Separability
of an extension field implies that each element on it is separable and each
element is algebraic and is the root of a separable polynomial
$f\in\mathbb{Q}(x)$. The separability of a polynomial can be verified if
$f(\alpha)$ and its formal derivative $f^{\prime}(\alpha)$ have no trivial
factors in common.

An important invariant to classify the Galois group is the discriminant, which
is defined as follows. Let $f\in\mathbb{Q}(x)$, be a degree n polynomial with
roots $\alpha_{i}$, the discriminant is defined as

$\begin{split}\text{disc}\,f=\prod_{i<j}^{n}\left(\alpha_{j}-\alpha_{i}\right)^{2}\,.\end{split}$
(102)

Thus, $\text{disc}\,f(x)$ is nonzero if $f(x)$ is separable (no repeated
roots). Thus, for a separable polynomial $f$, $\text{disc}\,f$ is a symmetric
polynomial in the the roots $\alpha_{i}$, and in consequence is fixed by the
Galois group of $f(x)$.

The discriminant of $f(x)$ is useful in reducing the possible Galois groups,
for instance, if $\text{disc}\,f$ is a square in $\mathbb{Q}$ then the Galois
group of $f(x)$ is a subset of the alternating group $A_{n}$. For instance, in
the case of an irreducible degree 3 polynomial the Galois group is uniquely
determined by this criterium since there are only two transitive subgroups of
$S_{3}$.

Now, let us define the fixed subfields under the K-automorphisms as

$\begin{split}L^{H}=\\{x\in L:\forall_{\sigma\in H}\sigma(x)=x\\}\end{split}$
(103)

as well as the fixed elements of the Galois group

$\begin{split}G(L/M)=\\{\sigma\in G(L/K):\forall_{x\in
M}\sigma(x)=x\\}\end{split}$ (104)

where the Galois group of an irreducible polynomial over the rationals of
degree n is a transitive subgroup of the symmetric group $\text{S}_{n}$.
Namely, for any two roots of the polynomial f, there exists an element of the
Galois group such that $\gamma(a)=b$. Thus, this restriction on the possible
subgroups reduces considerably the possibilities to take into account. We
shall employ the Eisenstein criterium to test the irreducibility of the
polynomials over $\mathbb{Q}$. For this criterium, let
$f(x)=\sum_{i=1}^{n}a_{i}x^{i}$ be a polynomial over $\mathbb{Z}$, the
polynomial shall be irreducible over $\mathbb{Q}$ if there is a prime q such
that i) $q$ does not divide $a_{n}$, ii) $q^{2}$ does not divide $a_{0}$ and
iii) $q$ divides all $a_{i}$ for $i<n$.

To look for the Galois group, in the following, we shall employ the algorithm
proposed by Stauduhar stauduhar1973determination , which is adequate for monic
irreducible polynomials over $\mathbb{Z}$. This is not a limit to the
algorithm since the polynomials that are typically involved in a tree-level
vacuum in string theory are non-monic polynomials over $\mathbb{Z}$, due to
the flux quantization conditions. The non-monic polynomial can be easily
transformed into a monic polynomial by taking the transformation

$f(\phi)\rightarrow a_{n}^{n}f\left(\frac{\phi}{a}\right)\,,$ (105)

where $a_{n}$ is the coefficient associated with the element of a higher power
in the polynomial and $f(\phi)\in\mathbb{Q}\left[\phi\right]$ such that it
preserves the splitting field. The Galois group is computed through invariants
derived from the structure of the groups. In the cases studied below, we
consider the G-relative H-invariants given that $H<G<\text{S}_{n}$, thus a
polynomial $F$ is G-relative H-invariant if its stabilizer

$\text{Stab}_{G}\,F=\\{\sigma\in G|F^{\sigma}=F\\}$ (106)

equals H. Furthermore, if $\text{Gal}(f)\leq G$ where $\text{Gal}(f)$ acts on
the roots $\alpha_{i}$, the relative resolvent is given by

$R_{F}(y)=\prod_{\sigma\in
G//H}\left(y-F^{\sigma}(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})\right)$ (107)

where $G//H$ is the set of representatives of right cosets $H\sigma$555To
compute the right cosets of $H\subset G$, it is selected an element of $g\in
G$ which is not contained in $H$ and $Hg$ is computed for each element. and

$F=\sum_{\sigma\in G}x_{\sigma(1)}x^{2}_{\sigma(2)}\ldots
x_{\sigma(n)}^{n}\,.$ (108)

is a $S_{n}$-relative H-invariant polynomial. However, these polynomials are
usually computationally too expensive, since each term needs $n-2$
multiplications, thus in the following the trimmed invariants proposed by
GEISSLER2000653  shall be employed.

The computation of the Galois group is based upon identifying the transitive
subgroup of $\text{S}_{n}$ by traversing the lattice of subgroups down to the
Galois group of the polynomial. To check the lattice subgroups, we shall
employ de decision tree together with the trimmed invariant polynomials
proposed in wildsmithcontribution . In this manner, the Galois group is found
independently of the numerical value for the fluxes. Besides, the Galois group
is checked against a specific selection of fluxes and the classification
provided by Magma.

Magma calculates the Galois group through the algorithm proposed by Klüners
and Geissler, which goes as follows. For an irreducible polynomial $f$ the
algorithm starts by factoring $f$ modulo several primes $p$ to look for a
prime such that the least common multiple of the factors is not quite large
keeping $f$ square-free. If the Galois group is in $\text{S}_{n}$ or
$\text{A}_{n}$, the group can be detected from the degrees of the modulo
p-factors. If the Galois group is not $\text{S}_{n}$ or $\text{A}_{n}$ the
next step is to calculate the roots of the polynomial up to high precision.
Numerically, this step is carried on by employing taking an approximation of
the roots in the p-adic field, however, for the present paper, the roots of
the polynomial can be written in an explicit manner depending on the NS or RR
fluxes for the case of solvable Galois groups.

Once the roots are known, the next steps require the knowledge of a suitable
starting group to start the Stauduhar iteration. An obvious starting group is
$\text{S}_{n}$, however, this starting group is not always the best choice,
since the computation of the $\text{S}_{n}$-relative H-invariants could become
computationally quite expensive. Instead, to determine the starting group, it
is better to compute all subfields of the stem field
$E=\mathbb{Q}\left[x\right]/f$ of $f$, the corresponding block systems and the
largest transitive group G admitting those blocks.

The third step is the iteration of the Stauduhar test. As a result, we have a
chain of subgroups with the properties described above; hence, we apply those
techniques to verify the result.

On the other hand, for reducible polynomials the Galois group of a compositum
polynomial $f=f_{i}f_{j}$ and for Galois groups $G_{1}(f_{1})$ and
$G_{2}(f_{2})$, the total Galois group is isomorphic to a group $G\in
G_{1}\times G_{2}$ awtrey2017determining ; SUTHERLAND201573 . Besides, the
Galois groups of the factors shall be invariant of $G$.

###  8.2 Decision trees

The decision tree for $S_{4}$ wildsmithcontribution  (for more complicated
decision trees check the reference). On the decision tree, the arrows at the
left indicate the case where the answer is positive. In consequence, the
dashed lines imply only subgroups of $A_{4}$.

$\sqrt{\Delta(f)}$ $D_{4}$ $D_{4}$ $V_{4}$ $A_{4}$ $Z_{4}$ $S_{4}$ $Z_{4}$
$D_{4}$ Figure 8:  Decision tree for $S_{4}$

For this particular case, the search for the Galois group starts by checking
if the discriminant is a perfect square. If it is not a perfect square, then
the decision tree leads to check if the invariant polynomial corresponding to
the subgroup $D_{4}$ contains any integer root, in case the invariant
polynomial contains an integer root, then the roots are orderer and the
invariant polynomial for the $Z_{4}$ subgroup is checked for invariant roots,
and the algorithm finishes by checking the invariant polynomial of either
$Z_{4}$ or $D_{4}$ depending if the invariant polynomial contains or not an
integer value.

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