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.
49 questions from the last 365 days
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On the form of algebraic numbers belonging to a specific field extension
Let $m>1$ be an integer and set $\theta=10^{-1/m}$. For a $\gamma\in \mathbb{Q}(\theta)$, there exists $a_0,\ldots,a_{m-1}\in \mathbb{Q}$ such that
$$
\gamma=a_0+a_1\theta+\cdots+a_{m-1}\theta^{m-1}...
- 515
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100
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Is there a (simple) criterion for membership to the base field of an inseparable extension?
Let $F$ be a field, let $f \in F[x]$, let $E$ be the splitting field of $f$, and let $e \in E$ be written in terms of the roots of $f$.
I'm looking for a simple way to establish if $e \in F$.
If $E/F$ ...
- 157
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80
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Galois group of shimura varieties with different level structure
Let $(G,X)$ be a shimura data, and $K$ an open compact neat subgroup of $G(\mathbb A_f)$. Suppose $K'\subset K$ is open and normal, then, I see in many references that the finite etale cover $Sh(G,X)_{...
- 775
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Reference Request: Pushforward of $\pi_1$ along a covering map and the Galois group
Let $f \colon (Y,y_0) \to (X,x_0)$ be a finite-to-one pointed covering map. The pushforward gives an inclusion $f_* \colon \pi_1(Y,y_0) \to \pi_1(X,x_0)$. If we take the universal cover $\widetilde{X}$...
- 949
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Quadratic forms with the same roots over GF(2) for low rank problems
Let $Q_1(x)=x^TA_1x$ and $Q_2(x)=x^TA_2x$ with $x\in GF(2)^n$, $A_i\in GF(2)^{n\times n}, i \in \{1, 2\}$. If $rank(A_1)=rank(A_2)=2$, is it possible that $Q_1(x)$ and $Q_2(x)$ can have the same roots ...
- 63
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Number of roots of a quadratic form over GF(2)
If $Q(x) = x^T A x$ with $x \in GF(2)^n$ and $A \in GF(2)^{n \times n}$, is there a way to find how many roots $Q(x)$ has based on any properties of $A$ (e.g., rank, etc.)?
- 63
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Subgroups that conjugate-cover the ambient group
Let $G$ be a finite group, and suppose that a set of proper subgroups $H_1,\dotsc,H_n$ satisfy $G=\bigcup_{g\in G}\bigcup_{i=1}^nH_i^g$, where $H_i^g$ is the conjugate of $H_i$ by $g$. In this case, ...
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Which elements in $\mathrm{Aut}(\widehat{F_2})$ preserve the procyclic subgroup generated by the commutator $c=[a,b]$?
Let $F_2$ denote the free group over two generators $a,b$, and we denote $c=[a,b]$ as the commutator. It is well-known that any automorphism $\psi$ of $F_2$ preserves the conjugacy class of the ...
- 605
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Classification of Étale algebras without Galois theory and then deducing Galois theory
In Milne's Galois theory notes — chapter 8, quoted below, he remarks that it is possible to classify étale algebras without using Galois theory then deduce Galois theory and he will explain this ...
- 253
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Beyond the Bring Radical: What is known about "generating radicals" for roots of polynomials of a given degree?
Famously, there is no general solution by radicals to find roots of polynomials (real, say) with degree $d\geq 5$. Somewhat less famously, there is a general solution[?] in degree $5$ using the so-...
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Is it possible to construct algebraic numbers from $\mathbb{Q}$ without using polynomials? [closed]
In $\mathbb{N}$, we can define an equivalence relation on the Cartesian product $\mathbb{N}^2$ as $(a,b) \sim (c,d)$ if and only if $a + d = b + c$. Then, the quotient set $\mathbb{N}^2 / \sim$ is ...
- 29
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$f(x)\bmod p$ and decomposition of prime ideals
While reading Serre's beautiful book Lectures on $N_X(p)$, I thought of a related question.
Let $f(x)\in \mathbb{Z}[x]$ be a monic irreducible polynomial with integer coefficients. Let $K$ be the ...
- 709
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Field extensions and completions at possibly infinite places
In Serre's Corps Locaux, Chapter 2 §3, is presented a classical proof. We are in an "ABKL" setup, where $K/L$ is finite, $A$ is Dedekind, $B$ is the integral closure of $A$ and $B$ is $A$-...
- 253
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inverse Galois problem on cyclic groups
It is known that the splitting field of $x^{p^n}-x$ over $\mathbb{F}_p$ is $\mathbf{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)\cong\mathbb{Z}/n\mathbb{Z}$ and the splitting field of $\Phi_n(x)$ over $\mathbb{...
- 11
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Kernel of inflation of $H^2$ in Galois cohomology
For a finite set $T$ of primes let $G_T$ denote the Galois group of the maximal extension $K_T$ of $\mathbb{Q}$ which is unramified outside $T$.
Let $S$ denote a finite set of primes and let $S' = S\...
- 2,907
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If $E_\text{sep}/F$ is normal, then must $E/F$ be normal?
This question has been asked in Math.StackExchange (see here) for more than a week and I even put a bounty on it. But still it hasn't been correctly answered (the current answer there was written by ...
- 452
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Is the Galois closure of a $p$-adic Lie group extension also a $p$-adic Lie group extension?
Let $p$ be a prime. Let $K$ be a number field and $L/K$ be an infinite extension which is not necessarily Galois. Suppose that the automorphism group $\text{Aut}(L/K)$ of $L/K$ is $p$-adic analytic, i....
- 667
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Galois groups of truncated $\cosh(x)$ Taylor polynomials and related results?
(Part of this question was written with ChatGPT because english is not my native language).
I am currently translating my diploma thesis from 2010 in english and thought to think about the topic again:...
- 3,064
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On a functional equation of Mahler?
Recently, I was trying to introduce the concept of natural boundaries to a fellow math student, and what greater way to do this than using an example? In particular, I tried to use as an illustration, ...
- 131
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459
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Why are smooth irreducible representations of the Weil group finite dimensional?
I am trying to understand the proof in Bushnell and Henniart - The local Langlands conjecture for $\operatorname{GL}_2$ of the fact (§28.6, Lemma 1) that smooth irreducible representations of the Weil ...
- 367
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127
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Irrational elements can always be moved
Let $x_1,x_2,x_3,\ldots,x_n$ be the roots of a polynomial $P_n(x)$. Let $F$ be the field $\mathbb{Q}[x_1,x_2,x_3,\ldots,x_n]$, i. e. all the possible combinations of rational numbers with $x$'s.
It's ...
- 125
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125
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Semisimplicity of induced representation of a irreducible representation
This question occurs when I read this one.
Suppose $k$ is a field of char $0$ (not necessarily complex numbers, in my case it's $\mathbb Q_p$), and $G$ is a profinite group (in my case, $\operatorname{...
- 775
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93
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Existence of maximal totally ramified subextension
Suppose $K/\mathbb Q_p$ is a finite extension, with ramification index $e$ and inertia index $f$. I want to ask whether there exists a subextension $L/\mathbb Q_p$ totally ramified of degree $e$?
- 775
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Inverse Galois problem for G with a derived sequence of length 2
For a finite group G with a derived sequence of length 2, please tell me how to specifically construct a field that is a Galois extension of $\mathbb{Q}$ and whose Galois group is G.
I made some edits ...
- 41
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English translation of Borel-Serre's "Théorèmes de finitude en cohomologie galoisienne"?
Is there an English translation of this text, or at least some English language paper that proves the same results?
I especially need a proof of the following fact which is in this paper: Say $k$ is a ...
- 31
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Describing the primes with each cyclic decomposition group in a given finite Galois extension of $\mathbb Q$
$\newcommand{\Q}{{\mathbb Q}}
$Let $f\in \Q[x]$ be a polynomial,
and let $L/\Q$ be the finite Galois extension
obtaining by adjoining to $\Q$ all roots of $f$.
Magma knows how to compute $\Gamma:={\...
- 14.1k
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Is there a uniform family of polynomials $f_p(x) =x^2 + a(p)x + b(p)$ such that $f_p(x)\in \mathbb{Z}[x]$ is irreducible and irreducible mod $p$?
Let $p\in\mathbb{Z}$ be a positive prime number.
Is there a "uniform" family of polynomials $f_p(x) =x^2 + a(p)x + b(p)$ of degree two such that $f_p(x)\in \mathbb{Z}[x]$ is both irreducible ...
- 237
3
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Do these polynomials with a complex kind of ‘Vieta jumping’ exist for all $k$?
Inspired by a recent question about sequences defined by $s_{n+1}=s_n^2-s_{n-1}^2$, I started wondering whether non trivial real or complex cycles of any length $k\geqslant3$ fixed by such a sequence ...
- 13.4k
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Noether's Problem and the Inverse Problem on Galois Theory
For the sake of simplicity, assume the base field $k$ as having zero characteristic. I will discuss 4 different formulations of Noether's Problem.
version 1 - original Noether's problem: Let $G<S_n$...
- 3,318
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Is every complex linear algebraic group a differential Galois group?
Let $ G $ be a complex linear algebraic group. In other words, $ G $ is a subvariety of the space of $ n \times n $ complex matrices and $ G $ is a group under matrix multiplication.
Does there always ...
- 4,081
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determinantal ideal of sum of Galois conjugate matrices
Given $n$ matrices $A_i \in \mathbb{Z}^{m\times m}$. I am interested in the ideal $I_d(A)$ generated by the $d\times d$-minors of $A = \sum_{i=1}^n x_iA_i \in \mathbb{Z}[x_1, \dots , x_n]$.
The matrix ...
- 61
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When does sum of algebraically independent polynomial become dependent?
Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ where $f_n = g + h$. Suppose the sets $\{ f_1,...,f_{n-1},g \}$ and $\{ f_1,...,f_{n-1},h \}$ are algebraically independent then is there a ...
- 341
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Galois action on étale path torsors
TLDR: How is the Galois action on étale path torsors defined?
Let $X$ be a smooth proper scheme, over a field $k$, and let $x,y\in X(k)$ be a pair of points. Let $\pi_1^{\text{ét}}(\overline{X},\...
- 2,907
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Who was the first to prove that the automorphism group of a finite field is cyclic and is generated by the Frobenius automorphism?
$\DeclareMathOperator\Aut{Aut}$It is well-known that the automorphism group $\Aut(F)$ of a finite field $F$ of characteristic $p$ is cyclic of order $n$ where $|F|=p^n$. Moreover, the cyclic group $\...
- 41.8k
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Semidirect product in inverse Galois problem
Let $L/\mathbb{Q}$ (resp. $K/\mathbb{Q}$) be a Galois extension of rational number field $\mathbb{Q}$ with Galois group $P$ (resp. $H$) where $P$ is a second countable pro-$p$ group and $H$ is a ...
- 667
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Constructing a cyclic extension $L$ with given local behavior of a global field $K$ such that $L$ is normal over a subfield $F$ of $K$
Let $F$ be a global field without real places
(that is, a function field or a totally imaginary number field).
Let $K/F$ be a cyclic extension of degree $n$.
Let $S$ be a ${\rm Gal}(K/F)$-invariant ...
- 14.1k
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Trying to understand the topology of the Weil group for the local Langlands conjecture
I am trying to study the representation of the Weil group from the book "The Local Langlands Conjecture for $GL(2)$". I have some problem with the topology of this group.
Let $F$ be a non ...
- 367
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Normality in a tower of cyclic extensions of global fields, as in Artin-Tate
Let $L_0$ be a global field without real places, that is, a global function field or a totally imaginary number field,
and let $V_f(L_0)$ denote the set of finite (that is, non-archimedean) places of $...
- 14.1k
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Degree 6 Galois extension over $\mathbb{Q} $
Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...
- 923
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Class numbers in the unramified biquadratic extensions of number fields
Let $K/k$ be an unramified biquadratic extension of number fields (i.e., $\operatorname{Gal}(K/k)\simeq V_4$), and $k_1$, $k_2$ and $k_3$ its three intermediate fields. I know, in general, we can ...
- 65
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147
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Prime splitting in the division field of an elliptic curve
Let $E/\mathbb{Q}$ be an elliptic curve with good reduction at two distinct primes $p, \ell$. Suppose the mod $\ell$ Galois representation associated to $E$ is surjective. Let $K=\mathbb{Q}(E[\ell])$ ...
- 1,422
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2
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Understanding absolute Galois group from its representations
Background. A major theme of modern number theory is to study the absolute Galois group $\text{Gal}(\overline{\mathbb Q} / \mathbb Q)$. Galois representation theory attempts to understand $\text{Gal}(\...
- 903
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1
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296
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Distinct characters with the same character values, outer automorphisms and Galois conjugation
Given an (irreducible complex) character of a finite group the following three construction all yield another irreducible character of the same degree:
multiplying by a degree 1 character
applying an ...
- 4,081
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0
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190
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Distinguishing between prime factors of cubic discriminant and polynomial discriminant
Let $f(x)\in\mathbb{Q}[x]$ be an irrreducible cubic with root $\alpha$. Let $K=\mathbb{Q}(\alpha)$. There may be primes dividing $\text{disc}(f)$ that don't divide $\operatorname{disc}(K)$, so an ...
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What was the "stormy discussion" about differential Galois theory at IHES?
In Kazuo Okamoto and Yousuke Ohyama's paper "Mathematical works of Hiroshi Umemura", Annales de la faculté des sciences de Toulouse Mathématiques, XXIX, no. 5 (2020) pp. 1053-1062, there is ...
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639
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A question on algebraic independence
Let $f_1,f_2,\ldots,f_n, g \in \mathbb{F}_q[x_1,...,x_m]$. Assume that $f_1,\ldots,f_n$ vanish at $0$, so that $\mathbb{F}_q[[f_1,...,f_n]]$ is a subring of $\mathbb{F}_q[[x_1,...,x_n]]$. Suppose that ...
- 341
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1
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246
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How do "Kummer closures" of fields look?
Let $F$ be a field and $A$ a finite abelian group. You can ask: does the regular representation $F[A]$ of $A$ split as a direct sum of 1-dimensional representations? This is equivalent to the ...
- 54.6k
1
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0
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103
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Abelian extensions of the rationals
Let $E$ and $F$ be finite abelian extensions of $\mathbb{Q}$ such that $E\cap F=\mathbb{Q}$. (You could take $E$ and $F$ as cyclotomic fields if that makes my question easier.) Set $K:=EF$ (the field ...
- 385
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Can K$_3$ of finite fields be related to Teichmüller cocycles?
This is sort of a blind shot, but...
For a ring $R$, its third algebraic K-group is given by $\operatorname K_3(R)=H_3(\operatorname{St}(R))$.
To simplify matters, let $R$ be a finite field $\mathbb ...
- 18.4k