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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95 views

$p$-adic Hodge theoretic properties of global Galois representations via $\ell$-Frobenii

Let $G_{\mathbb{Q},S} = \mathrm{Gal}(\mathbb{Q}_S/\mathbb Q)$ where $\mathbb Q_S$ is the largest algebraic extension of $\mathbb Q$ unramified outside a finite set of places $S$. Then the union over $\...
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Reducibility of a cubic over a number field

Given an extension $K = \mathbb{Q}(\alpha)$ by a cubic polynomial $g(x)\in \mathbb{Q}[x]$ (not necessarily Galois extension) is there a criterion for a cubic polynomial $f(x) \in K[x]$ to be reducible ...
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117 views

Centralizer of the absolute Galois group of a number field

By this answer, we know that if $K/\mathbb{Q}_p$ is a finite extension, the centralizer of $G_K$ in $G_{\mathbb{Q}_p}$ is trivial. The argument there uses that the abelinization of $G_K$ is the pro-...
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308 views

A quantity associated to a field extension

Let $F\subset E$ be a field extension. So $E$ has a natural structure of $F$-vector space. A vector subspace $V\subset E$ is a special subspace if $F\subset V$ and $V$ is closed under the inverse ...
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219 views

Can every square root be represented as a linear combination on roots of unity? [duplicate]

Messing around, I noticed that $$\sqrt{2}=e^{i\pi /4}+e^{-i\pi /4}$$ $$\sqrt{3}=e^{i\pi /6}+e^{-i\pi /6}$$ and (even more surprisingly) $$\sqrt{5}=e^{2\pi i/5}-e^{4\pi i /5}-e^{6\pi i /5}+e^{8 \pi i /...
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Effective Hilbert's Irreducibility Theorem and Irreducibility of $f(x)+1$, $f(x)\in\mathbb{Q}[x]$ reducible

Take a reducible polynomial $f(x)\in\mathbb{Q}[x]$. I am interested in the question: is $f(x)+1$ irreducible over $\mathbb{Q}$? For $f (x) = (x −a_1) · · · (x −a_m)$ with distinct $a_1,\ldots, a_m\in \...
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286 views

perfect fields in positive characteristic

Let $k$ be an infinite perfect field in positive characteristic $p$, i.e. every element of $k$ is a $p$th power. I am interested in properties of finite fields that can be extended to $k$. For example:...
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277 views

Dedekind Zeta functions of Biquadratic fields

Let $F/ \mathbb{Q}$ be a biquadratic field of Galois group $C_2 \times C_2$. Then I know that the Dedekind Zeta function of $F$ can be factored into $L$-functions as; $$\zeta_F(s) = \zeta(s) L(s, \...
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An explicit isomorphism $K^\times/K^{\times p} \cong K^\flat/\wp K^\flat$ where $K \supset \mu_p$ is perfectoid field of mixed characteristic $(0, p)$

Let $K$ be a perfectoid field of mixed characteristic $(0, p)$, i.e. $K$ has characteristic $0$ but its residue field has characteristic $p$. Further suppose that $K$ contains all the $p$th roots of ...
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Is $\operatorname{Aut}(\mathcal{M})$ a fundamental group in Grothendieck's sense?

This question is a follow-up to Are there infinitely many L-rigs? and to Is an automorphic form of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ determined by its L-function?. I copy paste a deepl ...
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Field whose absolute Galois group is $\mathbb{Z}_p$

Let $L$ be a perfect field. Assume there is an algebraic closure $L\subset \overline{L}$ and a prime $p$ such that $\mathrm{Gal}(\overline{L}/L)\cong \mathbb{Z}_p$ as topological groups. Is there a ...
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Connecting different ways of constructing cubic extensions of $\mathbb{Q}$

There are at least two ways to construct cyclic cubic extensions of $\mathbb{Q}$ as explained below. (A third one is given in the answer to an earlier question). Given $A, B, C$ integers with $A\neq ...
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Conjecture about arcsin and $\sqrt{\quad}$

Let $r(a,b)$ be a rational number depending on $a,b$ and nonnegative. For every $b$ there is an $a$ such that $r(a,b)$ is not $0$. Let $C(a,b)$ be a squarefree positive integer depending on $b$ and ...
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1answer
645 views

Are absolute Galois groups condensed?

Let $k^{s}$ be a separable closure of a field $k$. Is $Gal(k^s/k)$ a condensed group in the sense of condensed mathematics? If condensed, is it always solid?
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What is the indecomposable decomposition of holomorphic differentials of an Artin-Schreier curve C as a Z/p-representation?

I am attempting to decompose the holomorphic differentials of an Artin-Schreier-Witt curve as a $\mathbb{Z}/p^n$-representation. This is done in Theorem 1 of Madan-Valentini Automorphisms and ...
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Permutation group with a nice lattice of block systems

Let $X$ be a finite set and $G$ be a transitive subgroup of the symmetric group on $X.$ Recall that a (complete) block system for this action is a partition of $X = B_1 \cup \cdots \cup B_k$ into ...
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How hard is it to find the first layer of this basic $\mathbb{Z}_p$-extension?

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 $\mathbb{Q}(\zeta_{p^2})$ such that $[\mathbb{Q}_1:\mathbb{...
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116 views

References for Hopf Galois module theory

I am a PhD student and I am really interested in Galois module theory, both in a "classical" and in a "nonclassical" sense. In the last months I have been reading about Hopf Galois ...
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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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Origin and context of adjunctions inducing equivalences between full subcategories

The following is well-known. Theorem. Let $F\dashv U$ be a pair of adjoint functors $$F\colon \mathcal C\to \mathcal D, \qquad U\colon \mathcal D\to\mathcal C$$ with unit $(\eta_A\colon A\to U(F(A)))_{...
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Motivic Galois correspondence

Is there a Galois correspondence in motivic Galois theory ? If so, is there a mathematical work on this correspondence that i can find on the net ? Thanks in advance for your help.
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Classifying twists for a general moduli problem

Suppose $\mathfrak X$ is a (Deligne-Mumford/Artin/...) stack and denote by $|\mathfrak X(k)|$ the set of isomorphism classes of it's $k$-valued point. Can we say something in general about the fibers ...
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1answer
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Upper bound for discriminant of Galois closure

In Lang's Algebraic Number Theory book he uses a certain bound on the discriminant of the Galois closure of a number field $K$ without proof stating that it is an easy exercise. Let $\tilde{K}$ be the ...
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Finitely generated subgroups of the absolute Galois group

Consider the absolute Galois group $\operatorname{Gal}(\overline{\mathbb{Q}} / \mathbb{Q})$. It seems to me that, in general, providing an "explicit" description of the elements of this ...
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Do algebraic completion/topological completion of fields always terminate? If so, are they unique?

Take the field $\mathbb{Q}$, If we complete it topologically with respect to the Euclidean norm, we get $\mathbb{R}$, then if we complete it algebraically, we get $\mathbb{C}$. On the other hand, the ...
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442 views

What are the main open problems in Group Theory and Galois Theory? [closed]

I’m really interested in doing a PhD and the subjects I enjoyed the most were Group Theory and Galois Theory. What are some open problems in these areas that would be suitable for a PhD? Is Galois ...
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1answer
87 views

Galois group acts by the $i$-th power of the Teichmuller character on $H$

In the second paragraph of Soulé - Perfect forms and the Vandiver conjecture, it is written that: For any natural integer $i \le p − 2$, let $C^{(i)}$ be the subgroup of $C$, where the Galois group of ...
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Representing finite sums of rational powers of 2

Let $X \subseteq \mathbb{R}$. Let $A$ and $B$ be finite subsets of $X$. The statement $$\sum_{a \in A} 2^a = \sum_{b \in B}2^b \iff A = B $$ is true if $X = \mathbb{N}$ or $X = \mathbb{Z}$; this ...
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What is the Galois group of one ultrapower over another ultrapower?

Let $F$ be a field, let $E$ be a field extension of $F$, and let $U$ be an ultrafilter. Then my question is, what is the relationship between the Galois groups $Gal(\Pi_U E/\Pi_U F)$ and $Gal(E/F)$? ...
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1answer
62 views

When is a infinite transcendence-degree rigid fields fixed by a finite extension?

A field is rigid if it has no nontrivial automorphisms. Let $F$ be a rigid field which has infinite transcendence degree over $\mathbb{Q}$, and let $E$ be a finite extension of $F$. Then my question ...
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Finite groups arising as Galois groups of maximal unramified extension of number fields

I was wondering if it is known for which number fields the maximal unramified (non-abelian) extension is of finite degree or do we know the finite groups that arise as the Galois groups of these ...
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Counter example of a radical extension that is not Galois/normal over $\mathbb{Q}(\omega)$?

Most proofs of Galois theorem stating that "an equation is solvable in radicals if and only if its Galois group is solvable," show the left to right direction by induction on the height of ...
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What is known about the algebraic completion of a monoid?

It is the monoid obtained by adjoining all solutions of polynomial equations. I'll demonstrate how to adjoin a single solution to a polynomial equation to a monoid: Let $W$ be a monoid and let $p(x)=q(...
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Possible Galois groups of residue fields of ramification points of a Galois branched cover of curves

Here's the version of the question I'm particularly interested in (though I'm also interested in other variants described near the end). Let $K$ be the function field of a smooth projective variety ...
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Can you define the absolute Galois groupoid in von Neumann–Bernays–Gödel set theory?

One difference between the absolute Galois groupoid of a field and the fundamental groupoid of a topological space is that for the former the set of objects you want to range over is not actually a ...
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Is this an explicit construction of a Hurwitz space with Galois group Z/p, p distinct branch points, and inertia group Z/(p-1)?

I am desperately confused and would like a sanity check that the following moduli space/stack is a Hurwitz space/stack. I would also appreciate any references on the topic of the explicit construction ...
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1answer
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Inverse Galois problem for non-Galois extensions

The inverse Galois problem asks whether every finite group appears as the Galois group of a Galois extension of the rational numbers. Is anything known about the anologous problem, where the ...
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Galois theory of periods of algebraic varieties PhD project

I finished my MMath at Exeter in July of this year and I would like to undertake a PhD at the interface of algebraic geometry, number theory and Galois theory. More specifically, I recently came ...
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Linear relation between polynomial roots

Consider an irreducible polynomial $P\in\mathbb{Q}[x]$ of degree $n$ whose second leading coefficient is $0$ and $\alpha_1,\dots,\alpha_n$ its $n$ distincts roots. I am interested on the problem of ...
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298 views

Minimal polynomial in $\mathbb Z[x]$ of seventh degree with given roots

I am looking for a seventh degree polynomial with integer coefficients, which has the following roots. $$x_1=2\left(\cos\frac{2\pi}{43}+\cos\frac{12\pi}{43}+\cos\frac{14\pi}{43}\right),$$ $$x_2=2\left(...
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Galois groups associated to matrices

When $A\in M_n(\mathbb{Q})$, we consider the pencil $A-xA^T$. Then $p_A(x)=\det(A-xA^T)$ is a self-reciprocal polynomial. $p_A$ can only be irreducible if $n=2p$ is even. Question: For every $p$, does ...
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1answer
94 views

Galois extensions of ring spectra and subextensions

In Galois extensions of structured ring spectra, Rognes introduces the notion of a faithful $G$-Galois extension of ring spectra. Let me recall what this means: We have a commutative ring spectrum $R$ ...
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Infinite separable extensions

Let $L/K$ be an infinite algebraic separable extension of fields. One assumes that the fields are embedded in an algebraic closure $\Omega$. Consider an element $\alpha$ of $\Omega$ separable over $L$....
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1answer
191 views

Field of definition/moduli through the monodromy

Let $Y$ be a smooth projective curve defined over a number field $K$, and let $B$ be a subset of $Y(K)$. It is known that the isomorphism class of a branched cover $f:X(\Bbb{C})\rightarrow Y(\Bbb{C})$ ...
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1answer
138 views

Reference book for Galois extension [closed]

I need a reference for field extension and Galois extension (like an introduction) please. Thank you.
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212 views

Etale cohomology and Kummer theory

If $K$ is a field and $n \geq 1$ is such that $n \in K^{\times}$, then $H^1_{et}(\mathrm{Spec}(K),\mu_n)=K^{\times} / (K^{\times})^n$. This is easy to prove, see for instance Tamme, Etale Cohomology, ...
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Doubt regarding invariance of discrete absolute value under automorphism

I have been reviewing some basic algebraic number theory and $p$-adic analysis, and the following thought crossed my mind: if $F/ \mathbb Q$ be a finite Galois extension, and $\eta$ be a $\mathbb Q$-...
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105 views

Concerning the Galois resolvent

Let $K$ be an algebraic number field, $f(x) = 0$ a separable algebraic equation over $K$ of degree $n \ge 2$, i.e. having only simple roots $x_1, \dotsc, x_n$, and $L \mathrel{:=} K[x_1, \dots, x_n]$ ...
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1answer
180 views

Polynomials for the alternating group $A_n$

It is my understanding that the polynomial $f_n(x)=x^n-1$ has the Galois Group $(\mathbb{Z}/n\mathbb{Z})^*$, group of units of order $\phi(n)$. In some sense, these are the "simplest" ...
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Jacobson-style Galois theory on perfect closure

Promoted from stack.exchange since I received no response: Jacobson developed a 'Galois' correspondence for purely inseperable extensions of exponent 1 (only consisting of pth roots) $K/k$, where he ...

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