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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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179 views

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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1answer
147 views

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 ...
2
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1answer
84 views

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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1answer
317 views

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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88 views

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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243 views

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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1answer
90 views

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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1answer
178 views

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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1answer
228 views

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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1answer
157 views

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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339 views

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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1answer
115 views

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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1answer
210 views

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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47 views

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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186 views

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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1answer
269 views

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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1answer
176 views

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 ...
2
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1answer
120 views

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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1answer
366 views

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 ...
2
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1answer
76 views

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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2answers
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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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42 views

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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1answer
138 views

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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203 views

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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1answer
140 views

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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1answer
142 views

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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1answer
542 views

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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1answer
70 views

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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2answers
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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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155 views

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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0answers
152 views

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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1answer
405 views

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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2answers
291 views

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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1answer
107 views

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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247 views

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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0answers
108 views

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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0answers
95 views

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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1answer
101 views

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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4answers
498 views

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 =\...