# Questions tagged [galois-theory]

Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is, in some sense, simpler and better understood.

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### Frobenius eigenvalues algebraic numbers

Let $X$ be a smooth projective variety over $\mathbf{F}_q$ and $\overline{X}$ its base change to $\overline{\mathbf{F}_q}$.
By Deligne’s Weil I, the eigenvalues of the geometric Frobenius acting on $...

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### English literature close to “Algébre et Théories Galoisiennes” by Régine and Adrien Douady

I'm currently working on my undergraduate dissertation. I'm working on covering sapces of Riemann surfaces so my supervisor asked me to read the book I mention in the title: "Algébre et Théories ...

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### Why Triangle of Mahonian numbers T(n,k) forms the rank of the vector space?

I am looking for an explanation of why Triangle of Mahonian numbers T(n,k) form the rank of the vector space $H^k(GL_n/B)$?
With respect to the property of Kendall-Mann numbers where the statement ...

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### Is $\text{PSL}_2(\mathbb{F}_{p^m})$ known to be a Galois group over $\mathbb{Q}$ for $m>1$?

Let $\mathbb{F}$ be a finite field of characteristic $p$, is it known that $\text{PSL}_2(\mathbb{F})$ can be realized as a Galois extension of $\mathbb{Q}$ for any/all cases when $\mathbb{F}$ is not $\...

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### Does there exist an unramified $PSL_2(\mathbb{F}_p)$ extension of a quadratic field $K$?

Does there exist an unramified $PSL_2(\mathbb{F}_p)$ extension of a quadratic field $K$ for $p\geq 5$?

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### Is the following variant of Shafarevich's theorem known?

Let $Q$ be a finite simple group which may be realized as the Galois group of some extension of $\mathbb{Q}$ (like for instance $PSL_2(\mathbb{F}_p)$ for $p\geq 5$, or the monster group) and let $G$ ...

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### Local factors determine Weil representations - proof of the Artin representation case

This post can be seen as a continuation of this post I created on MathOverflow.
I want to understand the proof of the following Theorem from "Euler Factors determine Weil Representations" by Tim and ...

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### Elliptic curves and $GL(2)$ Iwasawa theory

Let $E$ be a elliptic curve without complex multiplication over a number field $F$. Let $F_n=F[E_{p^n}]$ and $F_{\infty}=F[E_{p^\infty}]$. So by a well know theorem of Serre, the Galois group $Gal(F_\...

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### Correspondence Between First Galois Cohomology and Semilinear Actions Up to Isomorphism

I've been stuck for a while on Exercise 1.9 of Bjorn Poonen's "Rational Points on Varieties". We start with $L/K$ a finite Galois extension with Galois group $G$, some $r \in \mathbb{Z}_{\geq 0}$, and ...

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### Local factors determine Weil representations - proof of the cyclic case

I already created this post on Math Stack Exchange but I was not so sure if this question fits better here. If it is not, I want to apologize in advance and feel free to delete my post.
I want to ...

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### Classes of curves with “determinant-like operation”

Consider a motivating example:
Let $E\in \mathbb{Q}[y][x]$ be of degree $n=2$ (in $x$) and separable when viewed as a member of $\mathbb{Q}[x,y]$. Therefore we can calculate it's roots in $\mathbb{Q}[...

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### Relationships among constructions of fundamental group for schemes

There seem to be several constructions of fundamental group for schemes and stacks: by Grothendieck, Deligne, Nori, Noohi, Esnault-Hai, Vakil-Wickelgren, perhaps others as well. I am trying to ...

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### Interpolation of families of local fields

Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$...

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### “Algebraization" of $p$-adic fields

Part 1: a single finite place. Let $K$ be a finite extension of $\mathbf{Q}_p$.
Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion ...

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### Having a separable extension of degree $n$ implies having a Galois extension of degree $n$?

I would like an explanation for the fact stated in the title. To repeat:
Question: How does one prove that if a field has a separable extension of degree $n$, then it has a Galois extension of ...

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### Cyclic cubic extensions and Kummer theory

The Galois cohomology group $H^1(\mathbb{Q}, \mathbb{Z}/3\mathbb{Z})$ classifies cyclic cubic extensions $K/\mathbb{Q}$ (specifically: the non-trivial elements correspond to Galois cubic field ...

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### Image of the norm map for degree $3$ galois extension over $\mathbb{Q}$

I want to construct a cyclic division algebra of degree $3$ over some degree $3$ Galois extension $E$ of $\mathbb{Q}$. So the construction is as follows:
As a set $D=E\oplus uE \oplus u^2 E$ where $u$...

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### Are all real-closed subfields of $\overline{\mathbb{Q}}$ conjugate?

Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. The absolute galois group $G_\mathbb{Q}$ of $\mathbb{Q}$ acts on the set of real-closed subfields of $\overline{\mathbb{Q}}$.
...

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### The coefficients of an irreducible polynomial in cyclic extension of fields

Assume we have some base field $F$, whose characteristic is neither $2$ nor $3$. Moreover, assume there are two Galois field extensions, a cubic extension $K / E$ and quadratic extension $E / F$ such ...

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### Recognizing the Galois group from the field discriminant

Along the lines of the general question "How much does the discriminant of a number field reveal about the field?", I was wondering how often it happens that the discriminant of some number field ...

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### Shafarevich's theorem about solvable groups as Galois groups

I am seeking references to any proofs of Shafarevich's theorem about solvable groups being Galois groups.

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### Unique factorisation of prime geodesics?

In T. Sunada's 1985 paper ``Riemannian coverings and isospectral manifolds'', it is noted that for a compact Riemannian manifold $M$, prime (closed and non self-intersecting) geodesics behave like ...

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### Normal closure and separable elements

Let $K\subset E\subset\bar{K}$ be field extensions, $\bar{K}$ an algebraic closure of $K$. Denote $E_s$ the field of separable elements of $E$ over $K$, denote $\tilde{E}\subset\bar{K}$ the normal ...

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### Does the $G$-norm coincide with the ordinary norm for “quasi-$G$-Galois” extensions

Let $S$ be a commutative ring, let $G$ be a finite group acting on $S$ via automorphisms (not necessarily faithfully), and
let $R$ be a subring of $S$ consisting of elements fixed $G$.
The extension $...

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### Centraliser of an absolute Galois group

Let $K$ be a finite extension of $\mathbb{Q}_p$.
Is the centraliser of $\operatorname{Gal}(\overline{K}/K)$ in $\operatorname{Gal}(\overline{\mathbb{Q}_p} / \mathbb{Q}_p)$ trivial ?
If yes, how can I ...

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### Characters of the kernel of the norm map of an extension of local fields

Let $E$ be a quadratic extension of a local nonarchimedean field $F$ of characteristic zero (and odd residual characteristic). Let $\sigma$ be a generator of the Galois group $G = Gal(E/F)$. I'm ...

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### What is the dimension of the mathematical universe?

Forcing construction in set theory leads to a new understanding of the mathematical (multi)universe by providing a machinery through which one can construct new models of the universe from the ...

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### Sign preserving Galois automorphisms

I have an algebraic number $\alpha \in \mathbb{Q}(\zeta)$, where $\zeta^n = 1$ is a root of unity (not primitive) given as a linear combination of powers of $\zeta$, i.e, $\alpha = \sum_{i=1}^k a_i \...

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### Lie algebra of derivations for a transcendental field extension and intersection fields

Suppose that $L$ is a finite Galois extension of the field $K$.
If $L_1$ and $L_2$ are subfields of $L$ containing $K$ then $L_1\cap L_2=L^H$
where $H$ is the group generated by ${\rm Aut}_{L_1}(L)$ ...

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### Why does $E\otimes_KE\cong EG$ imply that Galois theory works?

This is a part of statement in the book I do not fully appreciate. Suppose $E/K$ is Galois extension and $G$ the galois group of $E/K$. $E[G]$ is the group ring formed by finite group $G$.
"It is ...

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### Is the intersection of ramification groups in upper numbering of a $p$-adic local field trivial?

Let $K$ be a $p$-adic local field, for example $\mathbb{Q}_p$. Let $G$ be the absolute Galois group of $K$, and let $G^v$($v\ge -1$) be the ramification groups in upper numbering, then is it true ...

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### Finding a cyclic cubic extension of a field

Let $K$ be a field and let $E/K$ be a Galois extension of degree 6 with $\text{Gal}(E/K) = S_3$, the symmetric group on 3 letters. Pick two different transpositions $s_1, s_2$ in $S_3$ (hence $s_1s_2$ ...

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### Galois groups of a family of equations

I investigated a family of polynomials given by for $n>1$:
$x^n+(x+1)^n+...+(x+n-1)^n=(x+n)^n$
And I found, with the help of Magma, that the Galois group is $S_n$ for $n$ up to a hundred. The ...

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### Inverse Galois problem for $2$-groups with an involution as complex conjugation

It is known that the inverse Galois problem for solvable groups was solved by Shafarevich. My question is the following: given $G$ a finite $2$-group and $s$ an element of order $2$ in $G$. Can we ...

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### Quadratic equation over a global field of characteristic 2

Let $F=\mathbb F_{2^n}(t)$, and let $f=x^2+ax+b\in F[x]$. Is there any necessary and sufficient condition for $f$, depending on its coefficients, to have a root in $F$? I'm not interested in finding ...

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### Is there a field with finitely many abelian extensions, that is neither separably closed nor real closed?

If $K$ has only finitely many Galois extensions, then $K$ must be either separably closed or real closed. Are there any other fields whose abelianizations are finite extensions (i.e. whose absolute ...

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### Reference request: Volume 2 of Abhyankar's lectures on algebra?

Abhyankar has a magnificent, if meandering (check them out if you want to see what I mean), set of lectures on algebra.
The description:
This book is a timely survey of much of the algebra ...

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### Field of constants of a Galois extension of function fields

Let $F$ be a finite Galois extension of the rational function field $\mathbb Q(x)$. Let $k$ be the field of constants of $F$, i.e., the algebraic closure of $\mathbb Q$ in $F$. Is $k$ necessarily a ...

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### Abel and Galois (and Arnold)

Question Is there a connection between Abel and Galois theories of polynomial equations?
Recall that for every polynomial $p(x)\in \mathbb{Q}[x]$ (say, without the free coefficient), Abel considered ...

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### Geometric fundamental group and algebraically closed residue field

my questions relates to the following talk of Tsuji:
https://www.youtube.com/watch?v=2brDj26phP0
At around 10:30 of the video, Tsuji is interrupted by a man stating that his construction does not ...

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### Vanishing sums of roots of unity

Lets say I have a multivariate polynomial $P(x_1, \ldots, x_n)$ of degree at most $d = d(n) = n^c$ for a fixed constant $c$. I know that the Kronecker map ($x_i \to x^{d^i}$) preserves zeroness/...

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### Interesting (combinatorial) actions of the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$

I have read many times that it is crucial to understand the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) =: L$, in so far that some have stated (was it Richard Taylor ?) that ...

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### Abelian group extensions

Let $K$ be an imaginary quadratic field and $E$ be an elliptic curve with CM by $\mathcal{O}_K$. Is there a way to see that $K(j(E), h(E[\mathfrak{p}]))/K$ is an Abelian extension for some $\mathfrak{...

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### When is a bilinear form equivalent to a trace form?

Associated to a finite, separable field extension $L/K$, there is a natural nondegenerate bilinear form, the trace form, defined by $$\langle x,y \rangle := \mathrm{Tr}_{L/K}(xy)$$
Now, given a ...

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### Realization of the primitive action of a wreath product in a Galois group

Let $f$ be a polynomial over a field $K$ of degree $n$ such that $f(x^2)$ is separable. Assume that the Galois group $G$ of (a splitting field of ) $f(x^2)$ is maximal, that is to say, the wreath ...

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### Question about Corollary 7.3 from Silverman's The Arithmetic of Elliptic Curves

I am trying to understand an argument of Corollary 7.3 from Silverman's The Arithmetic of Elliptic Curves. I am stuck and I would appreciate any explanations.
Let $E$ be an elliptic curve over $K$, ...

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### Reference request: Regarding the image of inertia group being a subgroup of Aut($\widetilde{E}$)

Let $E$ be an elliptic curve over $\mathbb{Q}_p$ with potential good reduction. I was told that if $F$ is the smallest Galois extension over $\mathbb{Q}_p$ such that $E$ has good reduction then the ...

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### Interpretation or application of this analog of minimal polynomial

Recently I was thinking about images of number field elements under a polynomial with coefficients in a smaller field, and I came across the following construction. It did not have the properties I ...

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### Galois Transcendental Field Extension has characteristic Zero

Let consider a Galois transcendental field extension $T/K$, therefore for each subextension $L$ of $T/K$ we have $T^{\operatorname{Aut}(T/L)} = L$.
My question is how to prove that this conditions ...

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### Why do some uniform polyhedra have a “conjugate” partner?

While browsing through a list of uniform polynohedra, I noticed that the square of the circumradius $R_m$ of the small snub icosicosidodecahedron ($U_{32}$) with unit edge lengths is,
$$R_{32}^2 =\...