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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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}...
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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 $E/F$ ...
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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)_{...
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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 $\widetilde{X}$...
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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.)?
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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 ...
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Measure on the places of $\bar{\mathbb Q}$
Consider the set $S$ of all places of $\mathbb Q$ (i.e. the set of all absolute values up to equivalence). Then we can consider $S$ as a measure space with the counting measure $\mu$. Therefore $\mu(\{...
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On Ramanujan's pi formula $\frac 1\pi=\sum_{k=0}^\infty\frac {(4k)!}{k!^4}\frac {Ak+B}{396^{4k}}$ and the solvable quintic $z^5-5z-396 = 0$?
(Updated with new information.)
I. Five eta quotients and the Monster?
Given Dedekind eta function $\eta(\tau)$, define the five eta quotients which in fact are the McKay-Thompson series 1A, 2A, 3A, ...
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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 ...
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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 ...
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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 ...
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Compass and straightedge construction of Poncelet polygons
Gauss–Wantzel theorem states that
A regular n-gon is constructible with straightedge and compass if and only if $n = 2^kp_1p_2...p_t$, where $p_i$'s are distinct Fermat primes (A Fermat prime is a ...
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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 ...
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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 $A$-...
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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{...
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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\...
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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 ...
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How hard is it to find the first layer of this basic $\mathbb{Z}_p$-extension?
$\DeclareMathOperator\Gal{Gal}$Let $p$ be a prime number and $\zeta_{p^n}$ be a primitive $p^n$-th root of unity. We know that there is a unique subfield $\mathbb{Q}_1$ of $\mathbb{Q}(\zeta_{p^2})$ ...
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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....
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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:...
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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, ...
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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 ...
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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 ...
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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{...
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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 $e$?
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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 ...
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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:={\...
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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 ...
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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 ...
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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 ...
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Can the unsolvability of quintics be seen in the geometry of the icosahedron?
Q1. Is it possible to somehow "see" the unsolvability of quintic polynomials
in the $A_5$ symmetries of the icosahedron (or dodecahedron)?
Perhaps this is too vague a question.
Q2. Are there ...
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$A_5$-extension of number fields unramified everywhere
So I was having tea with a colleague immensely more talented than myself and we were discussing his teaching algebraic number theory. He told me that he had given a few examples of abelian and ...
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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 $G<S_n$...
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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 ...
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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 ...
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How to solve a quadratic equation in characteristic 2 ?
What do I do if I have to solve the usual quadratic equation $X^2+bX+c=0$ where $b,c$ are in a field of characteristic 2? As pointed in the comments, it can be reduced to $X^2+X+c=0$ with $c\neq 0$.
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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 ...
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Is the Leopoldt conjecture almost always true?
The famous Leopoldt conjecture asserts that for any number field $F$ and any prime $p$, the $p$-adic regulator of $F$ is nonzero. This is known to be equivalent to the vanishing of $H^2(G_{F'/F},\...
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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},\...
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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 ...
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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 $\...
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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 ...
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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 ...
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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 $...
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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\...
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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 ...
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What are the different theories that the motivic fundamental group attempts to unify?
I must preface by confessing complete ignorance in the subject. I've read introductory texts about the theory of motives, but I am certainly no expert.
In http://www.math.ias.edu/files/deligne/...
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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 $K=\mathbb{Q}(E[\ell])$ ...
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