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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How to understand Galois Representations as an undergraduate? [closed]

I am looking to doing my undergraduate thesis on Galois Representation Theory. I am currently taking a class in Galois Theory and learning Algebraic Geometry independently. Are there any resources ...
myr's user avatar
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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 $\...
Taras Banakh's user avatar
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2 votes
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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 ...
Mikhail Borovoi's user avatar
6 votes
1 answer
417 views

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 ...
Mario's user avatar
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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 $...
Mikhail Borovoi's user avatar
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165 views

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\...
Sky's user avatar
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4 votes
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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 ...
ayoub-chess's user avatar
2 votes
0 answers
126 views

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 $K=\mathbb{Q}(E[\ell])$ ...
Jeff H's user avatar
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17 votes
2 answers
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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}(\...
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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 ...
Ian Gershon Teixeira's user avatar
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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 ...
Nicolas Banks's user avatar
12 votes
0 answers
334 views

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 ...
Phil Harmsworth's user avatar
8 votes
1 answer
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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 ...
Rishabh Kothary's user avatar
4 votes
1 answer
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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 ...
Theo Johnson-Freyd's user avatar
1 vote
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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 ...
Steve Stahl's user avatar
2 votes
0 answers
114 views

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 ...
მამუკა ჯიბლაძე's user avatar
4 votes
0 answers
121 views

A normal extension of a number field of given degree that does not split over a given set of finite places

Let $K$ be a number field and $S$ be a finite set of non-archimedean places of $K$. Let $n>1$ be a natural number. Question. Does there exist a normal extension $L/K$ of degree $n$ such that $L\...
Mikhail Borovoi's user avatar
7 votes
0 answers
136 views

Non-abelian ray class fields for local fields

Let $K$ be a non-Archimedean local field. Then, thanks to work of Koch (when $K$ has positive characteristic) and Jannsen-Wingberg (when $K$ has characteristic zero, and odd residual characteristic) ...
Riccardo Pengo's user avatar
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0 answers
76 views

Squares in division ring extensions $\ell/k$ with $[\ell:k] = 2$

Let $k$ and $\ell$ be division rings such that $\ell$ contains $k$, and $[\ell : k] = 2$. When do I know that there is an element $a \in k$ such that $x^2 = a$ has solutions in $\ell$, but not in $k$?
THC's user avatar
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For a quadratic extension $E/K$, condition on a character $\chi:E^\times/E^{\times 2} \to C_2$ to give a $C_4$-extension $L/K$

Let $K$ be a finite extension of $\mathbb{Q}_2$, and let $E/K$ be a quadratic extension. By local class field theory, quadratic extensions $L/E$ correspond to quadratic characters $\chi:E^\times \to ...
Sebastian Monnet's user avatar
6 votes
0 answers
392 views

Extensions of p-adic number fields

Let $p$ be a prime number and $\mathbb{Q}_p$ be the $p-$adic rational field. Let $E/\mathbb{Q}_p$ be a fixed finite extension. On this site, I define a finite extension $F/E$ to be "good" if ...
Eric's user avatar
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2 votes
0 answers
106 views

Absolute Galois cohomology of function fields (of high-dimensional) varieties

What is known about the absolute Galois cohomology of function fields of varieties of dimension 2 or larger? Specifically, I am interested in multiplicative coefficients $\mathbb G_m$. I have seen ...
Sean Sanford's user avatar
1 vote
1 answer
95 views

Existence of a symmetric matrix satisfying certain irreducible conditions

Let $K$ be a field such that $ \mathrm{char}(K) \neq 2 $. Let $ p(x) $ be an arbitrary irreducible polynomial over $K$ of degree $n$. Using the rational canonical form, we can always construct an $ n ...
Sky's user avatar
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1 vote
0 answers
66 views

Galois connection for homeomorphisms

let $M = \mathbb{R^2}$ and $X = \{0\}$ and $G = Aut_X(M)$ the group of homeorphisms fixing $X$ (pointwise). Then we have, in analogy to classical Galois theory for field extensions, a Galois ...
Henry's user avatar
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3 votes
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207 views

Global class field theory and closure of unit groups

I'm looking for a reference for the following facts from global class field theory that I found without proofs. I will state them as questions, just in case I get the statement wrong. We fix $K$ a ...
Tim's user avatar
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3 answers
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A subgroup of index 2 for a solvable finite group transitively acting on a finite set of cardinality $2m$ where $m$ is odd

Let $G$ be a finite group acting transitively on the finite set $X=\{1,2,\dots, 2m\}$ of cardinality $2m\ge 6$ where $m$ is odd. Question 1. Is it true that $G$ always has a subgroup $H$ of index 2 ...
Mikhail Borovoi's user avatar
2 votes
1 answer
167 views

Is there work on differential Galois theory and infinite operators?

I'm curious about differential Galois theory and I've noticed that everything I read covers only finite order operators (e.g. $L = Y^n + a_{n-1} Y^{n-1} + \dots + a_0 Y$). Has there been any work on ...
PQ-'s user avatar
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7 votes
1 answer
411 views

A cyclic Galois extension over $ \mathbb{Q}(\omega)$

It is known that $\mathbb{Q}(\sqrt{-1})$ does not live in a cyclic Galois extension $L$ of $\mathbb{Q}$ of degree $4$. For example, the image of complex conjugation in $\mathrm{Gal}(L/\mathbb{Q}) = \...
Sky's user avatar
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0 votes
1 answer
141 views

Existence of linear disjointness between an algebraic number field and $p$-cyclotomic field over $ \mathbb{Q}$

Suppose $ K $ be an algebraic number field and $ n $ be an even integer. Is it possible to find atleast one $ p $ such that $ p\equiv 1( \text{mod}~ n)$ and $ \mathbb{Q}(\eta_p) $ is linearly disjoint ...
Sky's user avatar
  • 913
2 votes
1 answer
182 views

Chinese remainder theorem for composition

Let $f(x) \in F_p[x]$ and I know $f(x)$ modulo two polynomials $\phi_1(x)$ and $\phi_2(x)$. What sort of information about $f$ modulo the composition $\phi_1(\phi_2(x))$ can I recover? I'm looking ...
mtheorylord's user avatar
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0 answers
48 views

E=F(E^p) implies extension is algebraic

Let $F$ be a field of characteristic $p$ and $E/F$ be a finitely generated field extension such that $E=F(E^p)$. Then show that $E/F$ is algebraic. I have proven it in case $E$ is singly generated ...
Akash Yadav's user avatar
6 votes
1 answer
228 views

If degree $N$ polynomials always have a root, does there still exist an irreducible polynomial of degree $N+1$?

Given that $F$ is a field, let $F_n$ be the completion of $F$ with respect to roots of degree $n$ polynomials. For example this would make $\mathbb{Q}_2$ the field of (ruler and compass) constructible ...
Sam Forster's user avatar
1 vote
1 answer
296 views

Unramified extension over $ \mathbb{Q}_{p} $

Let $\mathbb{Q}_{p}$ be a p-adic field such that $ p \neq 2 $. We knew that for every $ n=2m $ there exists exactly one unramified extension $ K $ of $ \mathbb{Q}_{p} $ of degree $ n $, obtained by ...
Sky's user avatar
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6 votes
0 answers
248 views

What, if anything, do we hope and expect to understand about (classical) Galois groups?

I was reading Franz Lemmermeyer's introduction to Fermat's Last and Wiles' Theorem, where he states Galois representations $\rho_p : G_\mathbb Q\rightarrow GL_2(\mathbb Z_p)$ are used for studying ...
plm's user avatar
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6 votes
0 answers
338 views

Can Langlands correpondence be restated using topos?

Langlands correspondence describes an equivalence between Galois representations and automorphic representations under some conditions. Laurent Lafforgue applying Olivia Caramello thesis described in ...
jaylooker's user avatar
6 votes
2 answers
397 views

Good and bad reduction for twists of algebraic curves

Suppose we have two curves $C/\mathbb{Q}$ and $C'/\mathbb{Q}$ which are twists of each other i.e. they are isomorphic over a field extension $K/\mathbb{Q}$. Suppose that $C$ has good reduction at a ...
did's user avatar
  • 595
2 votes
1 answer
145 views

Automorphism of positive characteristic field

Suppose $K$ is a field with $\text{char}(K) \geq 0$. Let $L$ be a cyclic extension of $K$ with degree $2n$. We consider the generator $\sigma$ of the Galois group $\text{Gal}(L/K)$. I am interested in ...
Sky's user avatar
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3 votes
1 answer
695 views

irreducibility of the polynomial $ x^4 +1 $

Let $K$ be a field. We consider the polynomial $f(x) = x^4 + 1$. It is known that $f(x)$ is irreducible over $\mathbb{Q}$ but reducible over any finite field. Thus $ f(x)$ is reducible over any field $...
Sky's user avatar
  • 913
5 votes
1 answer
298 views

Galois action on algebraic K-theory of finite fields

This might be well-known to experts. I was just teaching a course where we went through some parts of Quillen's theorem computing the higher algebraic K-theory of finite fields. Denote by $\mathbb F_q$...
Andreas Thom's user avatar
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4 votes
1 answer
293 views

Fields in which $ -1 $ can't be written as sum of two square elements

We say a field $F$ has the property $*$ if the equation $x^2 + y^2=-1$ has no solution in $F$. For an example if $F$ is a subfield of real numbers then $F$ satisfies $*$. On the other hand if $ F $ is ...
Sky's user avatar
  • 913
0 votes
1 answer
109 views

Statistics of action of Galois group of number field on primes over unramified rational primes

Let $p \in \mathbb{Z}$ be prime and $K / \mathbb{Q}$ be a finite Galois extension. The Galois group $G$ of $K$ acts on the primes of $\mathcal{O}_K$ over $p$. Do we know any statistical information ...
Vik78's user avatar
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4 votes
0 answers
141 views

A variant of the inverse Galois problem

In Theorem I of Construction of maximal unramified p-extensions with prescribed Galois groups, it's proved that for any prime $p$ and any given finite $p$-group $G$, there exists a number field $F$ ...
stupid boy's user avatar
3 votes
1 answer
353 views

How often does algebraic-conjugacy imply conjugacy?

Recall that two elements $h_1,h_2$ of a finite group $G$ are called conjugate when $h_2 = gh_1 g^{-1}$ for some $g \in G$, and algebraic-conjugate when $h_2 = gh_1^a g^{-1}$ for some $a \in (\mathbb{Z}...
Theo Johnson-Freyd's user avatar
2 votes
1 answer
387 views

Algebraically closed fields with only finite orbits

The automorphism groups $\mathrm{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)$ of algebraic closures of finite fields $\mathbb{F}_q$ and the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\...
THC's user avatar
  • 4,313
10 votes
0 answers
220 views

If $H$ is a quotient of $G$, does there exist an $H$-extension of $\mathbb{Q}$ not contained in a $G$-extension?

Let $\phi\colon G\rightarrow H$ be a surjective homomorphism between finite groups. Assume that $\phi$ is not split, in other words there exists no homomorphism $\sigma\colon H\rightarrow G$ such that ...
Jef's user avatar
  • 949
0 votes
1 answer
135 views

Not containing $ 2^{i}$-th primitive roots of unity in cyclic galois extension of number field

Suppose $K$ is an algebraic number field. We want to determine whether, for all even $n$, there always exists a cyclic Galois extension $L$ of degree $n$ over $K$ such that the intermediate field $K_2$...
Sky's user avatar
  • 913
1 vote
1 answer
167 views

Norm of $2^{i}$-th primitive root

Let $ K $ be finite degree extension of $ \mathbb{Q} $ such that $ -1 $ is not a square in $ K $. Let $ L = \frac{K[x]}{\langle x^2 +1\rangle}$. Thus every element of $ L $ is of the form $ a + ib $ ...
Sky's user avatar
  • 913
6 votes
2 answers
504 views

Splitting fields of degree 4 irreducible polynomials containing a fixed quadratic extension

Let $f_1(x)\in \mathbb{Z}[x]$ be a fixed irreducible degree 4 polynomial such that its splitting field $F_1$ is an $S_4$-Galois extension over $\mathbb{Q}$ and the discriminant of $F_1$ is of the form ...
debanjana's user avatar
  • 1,161
15 votes
1 answer
283 views

Is a generic genus $g \geq 7$ curve a solvable cover of $\mathbb{P}^1$?

Let $Y \to X$ be a finite branched cover of smooth projective curves over $\mathbb{C}$, so we get a finite extension $K(Y)/K(X)$ where $K(\ )$ is the field of meromorphic functions. Say that $Y \to X$ ...
David E Speyer's user avatar
10 votes
1 answer
433 views

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 $z^5-5z-396 = 0$?

I. Four quintics? The general quintic can be transformed in radicals to at least three one-parameter forms. For simplicity, assume this free parameter to be some generic "alpha". Hence, $$x^...
Tito Piezas III's user avatar

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