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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Decomposition of the Galois group of the $m$-th division field of an elliptic curve with CM into a direct product of Galois groups

Let $E/\mathbb{Q}$ be an elliptic curve with CM from an imaginary quadratic field $K$. Let $K(E[m])$ denote $m$-th division field (number field obtained by adjoining the coordinates of the $m$-torsion ...
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25 votes
1 answer
1k views

A simple proof of the fundamental theorem of Galois theory

Update. It's now on the arXiv. Some time ago I found my "own" proof of the fundamental theorem of Galois theory. You can find a pdf with the proof (link removed, see arXiv). It is quite ...
2 votes
1 answer
152 views

A question about unramified quadratic extension of number field

Is there any condition over a number field $K$ for an unramified quadratic extension of $K$ to admit an embedding into an unramified cyclic extension of degree 4 of $K$?
2 votes
0 answers
114 views

Can we say anything about the zeros and Galois group of the polynomial $(x^p-a)^{p^2}-p^{p^2+1}x+p^{p^2} a=0$?

Let $p$ be an odd prime number and $\mathbb Q_p$ be the $p$-adic number field. Let $K=\mathbb Q_p(a)$ be the extension by $a=p^{\frac{p^2+1}{p^3-1}}$. Consider the polynomial $f(x)=(x^p-a)^{p^2}-p^{p^...
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3 votes
1 answer
112 views

Is there a theory of "elementary closed form solution" at the operator level for differential equations?

We begin by considering the usual general first order linear equation of the form $$ a_0 y' + a_1 y + a_2 = 0 $$ Where $a_i,y \in \mathbb{C} \rightarrow \mathbb{C}$. Now it's well known from everyone'...
5 votes
1 answer
266 views

When is $\mathbb{Q}(\sqrt{p+\sqrt{p}})$ a Galois extension of $\mathbb{Q}$?

When is $\mathbb{Q}(\sqrt{p+\sqrt{p}})$ a Galois extension of $\mathbb{Q}$? I was motivated by the question that $\mathbb{Q}(\sqrt{5+\sqrt{5}})$ is a Galois extension of $\mathbb{Q}$. Here is a rough ...
2 votes
0 answers
56 views

Galois cohomology with coefficients in the integers of the Lubin-Tate extension

Let $K$ be a $p$-adic local field, and $L$ the Lubin-Tate extension obtained from $K$ by attaching roots of some Lubin-Tate formal $\mathcal{O}_{K}$-module with $Gal(L/K) \simeq \mathcal{O}_{K}^{\...
4 votes
1 answer
207 views

What are the jumps in the ramification filtration of the absolute Galois group of a local field?

Let $k$ be a (complete) discretely valued field and $\ell$ a Galois extension of $k$, possibly infinite. The Galois group $\Gamma=\text{Gal}(\ell/k)$ of $\ell$ over $k$ admits a descreasing, $\mathbb ...
5 votes
1 answer
310 views

Cycle type in Galois group from ramified primes

Let $P \in \mathbb Z[X]$ be monic, separable, of degree $d$, $K$ its splitting field over $\mathbb Q$ and $G$ the Galois group of $K$ over $\mathbb Q$. Now, let $p$ be a prime number unramified in $K$....
6 votes
0 answers
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How to construct this non-geometric mod $p$ Galois representation?

Let $ K $ be a totally real cubic field. I am concerned with the following kind of Inverse Galois Problem: Is there a prime $ p>5 $ and a continuous representation $ \bar{\rho}:G_{K}\to {\rm GL}_{...
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1 vote
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Reference request: arithmetical implications of an ambient Galois extension

This is a cross-post of a question I asked on StackExchange. See there for further details. Let $L/K$ be a Galois extension of algebraic number fields of finite degree over $\mathbb{Q}$, with group $G$...
5 votes
0 answers
117 views

Can we deduce that both $\alpha$ and $\beta$ are algebraic over F if $F[\alpha,\beta]=F(\alpha,\beta)$

So after reading David Cox. book on Galois Theory, it can be shown that “Given a field F, $F[\alpha]=F(\alpha)$ if and only if $\alpha$ is algebraic over F” (See Prop. 4.1.14). Later, it was mentioned ...
-2 votes
1 answer
205 views

In Galois theory, why solvable groups must have their quotient groups be Abelian? [closed]

The definition of solvable groups can be regarded as two constraints, one is that there must be a sequence of normal subgroups, and the other is that the quotient groups between these sequences are ...
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Colimits in the category of suplattices

I want to compute coequalizers in the category $\mathcal{S}up$ of complete lattices and $\bigvee$-preserving maps. One way (I think?) is to use the dual equivalence $$ \mathcal{Sup} \leftrightarrows \...
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8 votes
1 answer
266 views

The distribution of certain Galois groups

Let $f(x)$ be a polynomial of degree $d$ with integer coefficients. Let $G_p^+$ be the Galois group of the polynomial $f(x)-y$ over $\overline{\mathbb{F}}_p(y)$ and $G_p$ be the Galois group of the ...
6 votes
2 answers
278 views

Algebraic numbers which prescribed degree which does not belong to some fields

In my research it would be great if the following result is valid. In what follows, $\overline{\mathbb{Q}}$, $\overline{\mathbb{Q}}_n$ and $\overline{\mathbb{Q}}_{<n}$ denotes the set of algebraic ...
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3 votes
0 answers
63 views

The cyclic analogue of the gonality of the superelliptic curve $s^n = t^m + 1$

For naturals $n$, $m > 1$ consider the superelliptic curve $C\!: s^n = t^m + 1$, for simplicity, over an algebraically closed field of zero characteristic or large characteristic $p \nmid n$, $m$. ...
3 votes
1 answer
300 views

Completion of infinite degree extension of perfectoid fields is perfectoid?

Is completion of infinite degree extension of perfectoid fields perfectoid ? It is known that finite extension of perfectoid fields is also perftoid from tilting correspondence, but what about ...
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5 votes
2 answers
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Absolute Galois group, number theory and the Axiom of Choice

Richard Taylor once explained his research (in number theory) in a very simple way: understanding the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. It is known that in ...
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2 votes
0 answers
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Embeddings and images of number fields in $\mathbb{C}$ [closed]

Let $\mathbb{Q}(\alpha)$ be a number field, and suppose that $[\mathbb{Q}(\alpha) : \mathbb{Q} ] = m$. Then there are precisely $m$ different embeddings of $\mathbb{Q}(\alpha)$ into $\mathbb{C}$, ...
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2 votes
0 answers
175 views

Dessins d'enfants and the absolute Galois group

If I am not mistaken, Alexander Grothendieck introduced dessins d'enfants as they are known today, in order to better understand the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}...
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0 votes
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Local field such that the value group of $K^\text{perf}$ ( perfect closure of $K$) is $\bigcup_{n\geqq1}(1/p^n) \Bbb{Z}$

Let $K$ be a local field of positive characteristic. I'm looking for a $K$ which satisfies the following condition. Value group of $K^\text{perf}$ (perfect closure of $K$) is $\bigcup_{n\geqq1}(1/p^n)...
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2 votes
1 answer
231 views

Can a general quintic be solved using inverse beta regularized function?

Tyma Gaidash has recently posted solutions to some quintics in terms of Inverse Beta Regularized function. He also found the closed form for the equation $\cos x=x$ using the same Inverse Beta ...
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5 votes
1 answer
189 views

Is it possible to solve sextic equations using the Fox H function?

Although the Kampé de Fériet function can solve the sextic equation, the details about it are shrouded in the fog of more than a century ago. In contrast, we know more about the Fox H function, and we ...
5 votes
1 answer
269 views

Rationality of field embeddings

After my earlier question question turned out to have a negative answer (Thank you to all respondents!), here is a more modest one. Both a positive answer and a counterexample would help my work. If ...
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7 votes
0 answers
133 views

Finding when a certain product in a cyclotomic field is equal to one

For the following, fix $m\geq 2$, and let $a_{0},\dots,a_{m-1}\in\mathbb{Z}$ be such that $\sum_{k=0}^{m-1}a_{k}=0$. I would like to find the exact conditions on the $a_{k}$ so that the following ...
6 votes
2 answers
269 views

Cancellation of irreducibility for Galois conjugates

Motivation: Take an algebraic number $\lambda$. In my research, I've stumbled upon the question in which cases the expression $\sum_{\sigma \in S} \sigma(\lambda)$, where $S$ is a subset of the field ...
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9 votes
2 answers
548 views

Non-trivial automorphisms and descent

In this expository paper by Low it says: Roughly speaking, a topos in the sense of Grothendieck is the category of sheaves on a kind of generalised space whose “points” may have non-trivial ...
4 votes
1 answer
353 views

On the solvability of the equation $ax^p+bx^{p-1}+cx+d=0$ by radicals

Joint with Qing-Hu Hou at Tianjin Univ., we seek for explicit criteria via coefficients for the solvability of an algebraic equation by radicals. In this direction, we formulate the following ...
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6 votes
2 answers
538 views

What is $\mathbb Q^{\mathrm{hypoab}}$?

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gal{Gal}\newcommand{\ab}{\mathrm{ab}}$Let $G(\mathbb Q) = \Gal(\overline{\mathbb Q} / \mathbb Q)$ be the absolute Galois group. It's well-known that ...
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3 votes
1 answer
280 views

A criterion for the equation $ax^n+bx+c=0$ not solvable by radicals via $a,b,c$ and $n$

Galois revealed that an algebraic equation $f(x)=0$ with coefficients in a field $K$ of zero characteristic is solvable by radicals if and only if the Galois group of $f(x)$ over $K$ is solvable. ...
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6 votes
1 answer
400 views

Is there a conjectured dependence on $n$ in van der Waerden's conjecture?

Bhargava 2021 proves van der Waerden's conjecture about Galois groups of random integer polynomials: over all $x^n + a_{n-1} x^{n-1} + \cdots + a_0 = 0$ with $a_k \in \{-H, \ldots, H\}$, the number of ...
5 votes
2 answers
122 views

Dihedral extension unramified at primes dividing order of group?

Consider dihedral Galois extensions $L/\mathbb{Q}$ of degree $n$ (and we know they exist thanks to Shafarevich), can we show there always exists an extension $L/\mathbb{Q}$ unramified at $p$, for all $...
2 votes
1 answer
289 views

Groupoid of points, shape and stratified shape of $\operatorname{Sh} (X_\text{pro-ét})$

$\DeclareMathOperator\Sh{Sh}\DeclareMathOperator\Pt{Pt}$Maybe this is well-known or even a stupid misunderstanding of something very basic. It's well-known that the groupoid of points (i.e., groupoid ...
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2 votes
0 answers
107 views

A dimension problem related to an abelian simple extension of a field

$\DeclareMathOperator\Imm{Im}$Let $K=F(\alpha)$ be an abelian extension of $F$ and let $\sigma$ be a map (could be any map) from $K^\times$ (the multiplicative group of $K$) to itself. Define an $F$-...
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2 votes
0 answers
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Number fields with given discriminant

In the special case of number fields that are the splitting fields of irreducible polynomials of degree 5 or 6 with Symmetric Galois group (so degree 120 or 720 over Q), is there a good upper bound on ...
5 votes
1 answer
289 views

Infinite Galois equivalence

I am having some trouble trying to understand the proof of Theorem 7.2.5 in Bhatt and Scholze's paper The pro-étale topology for schemes. Specifically, I don't quite understand why it was necessary to ...
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5 votes
0 answers
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Can there be non-isomorphic fundamental groups of equivalent Galois categories?

It is known that if $(C, F)$ is a Galois category then there exists an equivalence $C \cong \pi_1(C, F)-FinSets$ between $C$ and the category of finite sets with continuous actions of $\pi_1(C, F) := ...
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4 votes
0 answers
117 views

Proving that the fundamental group of a finite Galois category is profinite

This is sort of a cross-posting of this question of mine over on math.SE. Suppose that I am given a finite Galois category $(\mathcal{G}, F)$, i.e. a Galois category in the sense of SGA 1. One of the ...
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6 votes
0 answers
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Reference request: projectivity of the absolute Galois group of $\mathbb{Q}^{\mathrm{ab}}$

$\newcommand{\ab}{\mathrm{ab}}$Let $\mathbb{Q}^{\ab}$ denote the maximal abelian extension of $\mathbb{Q}$. I have heard the absolute Galois group of $\mathbb{Q}^{\ab}$ is projective (e.g. see this ...
3 votes
0 answers
145 views

What is the meaning of local inertia conjugation property?

In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030., we have: Abstract. Let $\widehat{G T}^{1}$ ...
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11 votes
2 answers
889 views

Polynomials for which roots can be expressed as polynomials in a single root

Classical Galois theory gives necessary and sufficient conditions for the roots of a polynomial in $k[x]$ to be expressible in terms of nested radicals of the coefficients. Suppose instead that a ...
9 votes
0 answers
269 views

What does Hilbert's 90 theorem tell us about Galois fixed points in projective space?

Consider the following statement: If $K\subseteq L$ is a Galois extension of fields with Galois group $G$ and $x \in \mathbb{P}^n(L)$ is such that $\sigma(x)=x$ for all $\sigma\in G$, then $x \in \...
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3 votes
0 answers
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Reference request: Radicial morphisms & Jacobson-Bourbaki correspondence

My name is Chemy (Przemysław). I am a PhD student at UvA (Amsterdam), and I work on projects related to foliations in algebraic geometry in positive characteristic. Therefore I am avidely reading two ...
5 votes
0 answers
287 views

To what extent are geometric methods being used to attack the inverse Galois problem?

My limited knowledge so far is that some groups have been constructed geometrically using the theory of covering spaces, then applying Hilbert irreducibility. Is there a deeper way in which inverse ...
2 votes
0 answers
95 views

Finite groups $G$ such that every Galois extension of group $G$ is totally real

This is an extended repost of the following question asked on MSE two weeks ago. I was playing with Galois extensions of small degree while preparing the final exam for my course in Galois theory, ...
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12 votes
2 answers
805 views

Topos-theoretic Galois theory

This page is an overview of some of the types of "Galois theories" there are. One of the most basic type is the fundamental theorem of covering spaces, which says, roughly, that for each ...
0 votes
0 answers
80 views

About the Galois group of product of two irreducible polynomials [duplicate]

If we have two ploynomials $p_1(t)$ and $p_2(t)$ over $Q$, and if them have no same root, suppose their Galois group are $G_1$ and $G_2$ respectively, if we can get the Galois group of $p_1p_2$ is ...
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2 votes
0 answers
194 views

Degree $8$ cyclic extension over $\mathbb{Q}$

Actually I am interested in degree $ 8 $ cyclic extension over $ \mathbb{Q} $. Let $ L $ be such extension. At first I was thinking to take basis as normal basis, as we can determine the galois group ...
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0 votes
1 answer
151 views

Does there exist a proper intermediate field between ℚ and ℚ̅ closed under taking nth roots? [closed]

Title says it all. Sorry my previous question was wrong; I see now; very stupid of me. So this is what I meant to ask. I am looking for a field extension of $\mathbb Q$, let's call it $K$, s.t. $K$ is ...

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