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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What are the possible sets of degrees of irreducible polynomials over a field?
Hopefully this is not too easy an exercise.
Let $F$ be a field. Let $I \subset \mathbb{N}$ be the set of all positive integers $d$ such that there exists an irreducible polynomial of degree $d$ over ...
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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 $K$....
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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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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 $G$...
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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 ...
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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 ...
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Degree of sum of algebraic numbers
This is an elementary question (coming from an undergraduate student) about algebraic numbers, to which I don't have a complete answer.
Let $a$ and $b$ be algebraic numbers, with respective degrees $...
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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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Cubic polynomials with "nice" roots, which can be expressed by trig functions of rational angles
Consider the cubic polynomial $x^3-ax+b$ for $a,b\in\mathbb N$.
It has three real roots which, by Cardano's formula, can of course be written in closed form using thirds of angles or cube roots of ...
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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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"Conjugacy rank" of two matrices over field extension
I have posted this elsewhere and got only a partial reply. I don't know whether this qualifies the question for an open-problem tag; if it does, please anyone insert it.
Let $L$ be a field, and $K$ a ...
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Galois groups vs. fundamental groups
In a recent blog post Terry Tao mentions in passing that:
"Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue ...
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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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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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What was Galois theory like before Emil Artin?
I read that the primitive element theorem for fields was fundamental in expositions of Galois theory before Emil Artin reformulated the subject. What are the differences between pre and post-Artin ...
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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 $p \nmid n$, $m$. ...
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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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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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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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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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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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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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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 ...
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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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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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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 ...
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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 ...
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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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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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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 ...
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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 $...
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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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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 ...
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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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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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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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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 ...
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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 ...
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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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What is the relation of the absolute Galois group and classical profinite groups?
Consider the absolute Galois group $G = \mathrm{Gal}(\overline{\mathbb{Q}}: \mathbb{Q})$ and $G_p = \mathrm{Gal}(\overline{\mathbb{Q}_p}: \mathbb{Q}_p)$.
Abelian class field theory gives us for the ...
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Why is differential Galois theory not widely used?
E.R. Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...
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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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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 ...
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Nonrepresentability by radicals and entire (or meromorphic) functions of algebraic functions
It is known that an algebraic function with non-solvable monodromy group can not be represented by radicals. Where can we find a detailed proof about the nonrepresentability by radicals and entire (or ...
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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 ...
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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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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 ...
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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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When the splitting fields of shifted generic polynomials are linearly disjoint?
Let me start by rigorously pose my question.
Let $K$ be an algebraically closed field of characteristic $2$, let $n$ be an even integer number, let $f(X) = X^n + T_1 X^{n-1} + \cdots + T_n$, be the ...
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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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