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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1answer
141 views

The upper bounds on rank $ 2 $ real matrices

Let $ A_{n}(F) $ be the collection of all skew-symmetric matrices over the field $ F $ ($\operatorname{char} F \neq 2 $). Let M be a subspace of $ A_{n}(F) $ such that all non zero elements have rank ...
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Simple proof of the equivalence between two definitions of étale

This question shouldn't be too hard to answer, but I'm looking for the most streamlined approach. Let $K$ be a field and let $L$ be a finite dimensional field extension of $K$. I am interested in two ...
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215 views

Restricting maps between strict henselisations

$\require{AMScd}$I am currently thinking about (strict) henselisations but I don't know too much literature about the topic. So I am wondering if there is a natural way to restrict maps between strict ...
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If $G=\mathsf{Aut}_k(F)$ acts on field $F$ algebraic over $k$ then do we have: orbit $G\alpha=\text{ roots of minimal polynomial of }\alpha$?

I posed this question on Math.Stackexchange (see here) but until now there was no response. This made me decide to give it a try here. Let $k\subseteq F$ denote an algebraic field extension and let $\...
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Galois extension of $ C_{k} $ field and irreducible polynomial over $ C_{k} $ field

A field is called a $ C_{k}$- field if every form i.e. every homogeneous polynomial of degree $ d $ in $ n > d^{k} $ indeterminates has a nontrivial zero. We know that any finite field is a $ C_{...
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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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1answer
75 views

Problem concerning about an $n$-subspace of $ A_{n}(F) $

Let $A_{n}(F) $ denote the $n \times n$ skew symmetric matrices over a finite field $F$. Suppose $n$ be even and $N$ be a subspace of $A_{n}(F) $. Now if all the non-zero matrices in $N$ are ...
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Existence of an irreducible polynomial that does not divide $x^n + a$

The question: Can one characterize all fields $K$ over which there exists an irreducible polynomial $f(x)$ that does not divide a polynomial of the form $x^n + a$? Examples: Such a polynomial clearly ...
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Index of norm $ 1 $ subgroup in a cyclic extension

Let $L/\mathbb{Q}$ be a cyclic galois extension of degree $ 2n $ and $\sigma $ be a generator of $\operatorname{Gal}(L/\mathbb Q)$. Let $ U $ be the collection of all norm $ 1 $ elements of $L^\times$...
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Solution of an equation in cyclotomic extension over $ \mathbb{Q} $ of degree $6$

Let us consider a primitive $7^{\text{th}}$ root of unity $\eta$. Then the minimal polynomial of $ \eta $ over $ \mathbb{Q} $ is $1 + \eta +.....+ \eta^{6}$. So the dimension of the $\mathbb{Q}$-...
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Resolvent is minimal polynomial for universal splitting algebra

Given a degree $n$ monic $f\in A[x]$ write $\mathrm{Split}_Af$ for its universal splitting algebra, constructed by taking the quotient of $A[x_1,\dots ,x_n]$ by the Vieta formulas. This is the initial ...
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Simultaneous Galois closure

For a finite separable extension $L/K$ of fields, there exists a Galois closure, which is a finite field extension $\tilde L/L/K$ where $\tilde L/K$ is Galois. (given by the compositum of $\sigma L$, ...
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When is the extension $L(S)/L$ Galois and totally ramified?

Let $L$ be a finite extension of the $p$-adic field $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_K$ with uniformizer $\pi$. Let us consider the polynomial ring $L[x_1,x_2,\dotsc,x_l]$ in $l$-...
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1answer
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Is there a measurable isomorphism ${\mathbb C}\to {\mathbb C}_p$?

Let $p$ be a prime and ${\mathbb C}_p$ be the completion of the algebraic closure $\overline{{\mathbb Q}_p}$. This field is isomorphic to $\mathbb C$. Both fields come with natural absolute values but ...
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1answer
162 views

Blow-ups of surfaces over a field

Let $S$ be a smooth projective surface of Picard rank $\rho(S)$ over a field $K$, and $\overline{S}$ its algebraic closure. Take a point $p\in\overline{S}$ and denote by $\overline{X}$ be blow-up of $\...
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1answer
262 views

Maximal common isotropic subspace for a finite family of skewforms

Let $V$ be a vector space of dimension $n$ over a field $F$. An alternating bilinear form $\alpha\colon V \times V \rightarrow F $ will be called a skewform. A subspace $W$ is isotropic for $\alpha$...
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207 views

Field extension generated by the roots of multivariate-polynomials

Let us consider a field $K$ of characteristic $0$. Then we know that any finite extension $L$ of $K$, which is a Galois extension as well, is produced the roots of a separable polynomial $f(x) \in K[x]...
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Does any cubic polynomial become reducible through composition with some quadratic?

What I mean to ask is this: given an irreducible cubic polynomial $P(X)\in \mathbb{Z}[X]$ is there always a quadratic $Q(X)\in \mathbb{Z}[X]$ such that $P(Q)$ is reducible (as a polynomial, and then ...
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On the characteristic polynomial of the Vandermonde matrix

Let $A_n$ be the $n \times n$-Vandermonde matrix (see for example https://en.wikipedia.org/wiki/Vandermonde_matrix )viewed as a matrix over the fraction field of the polynomial ring over a field $K$ (...
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Nef and pseudo-effective divisors over non algebraically closed fields

Let $X$ be a projective variety over a field $K$. As a consequence of Kleiman's criterion, when $K$ is algebraically closed, we have that if $D$ is a nef divisor on $X$ then $D$ is pseudo-effective. ...
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Inverse Galois problem on simple groups

Im trying to find a way to connect a possible solution of the inverse Galois problem on simple groups to a more general solution on any finite group. I've tryied to mess with the embedding problem for ...
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1answer
380 views

Original proof of Chebotarev's density theorem

As the title suggests, I am currently trying to understand Chebotarev's original proof of his density theorem, based on the proof in the appendix here. I am fully on-board with the cyclotomic ...
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1answer
354 views

On the Galois group of the compositions of polynomials

We reprint an old math SE question here (see https://math.stackexchange.com/questions/1241224/composition-of-polynomials-and-galois-theory): " Let $f(x)$ be a polynomial of degree $n$ over $\...
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How to show an invariant subfield of rational function field $\mathbb{Q}(x)$ under a certain group action is actually a simple extension? [duplicate]

Let $K=\mathbb{Q}(x)$ be the rational functions in one variable $x$ and let the automorphisms $\phi,\psi$ of $K$ be defined as $\phi(x)=-\frac{1}{x+1}$ and $\psi(x)=\frac{1}{x}$. Let $G$ be the group ...
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Simplification of links between idele class group and étale cohomology

I posted this question over on stack exchange and was told it would work better here. For interest I have been looking at links between class field theory and étale cohomology. Let $k$ be a global ...
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1answer
516 views

Subspaces of $ A_{n}(\mathbb {Q})$ in which all nonzero matrices are invertible

Let $A_{n}(\mathbb{Q}) $ denote the $n$ times $n$ skew symmetric matrices over the rational number field. Let $N$ be a subspace of $A_{n}(\mathbb{Q}) $. If all the non-zero matrices in $N$ are ...
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1answer
275 views

Galois module theory: from global to local

Let $L/\mathbb{Q}$ be a finite Galois extension with Galois group $G$. It is well known that the ring of integers $\mathcal{O}_L$ is free over its associated order $\mathfrak{A}_{L/\mathbb{Q}}=\{x\in \...
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188 views

The Galois resolvent in Lagrange

In Edwards' "Galois Theory" articles 29-31, the notion of Galois resolvent is motivated by a result of Lagrange (article 104 in his Réflexions sur la résolution algébrique des équations). ...
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$p$-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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160 views

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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1answer
137 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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2answers
342 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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1answer
280 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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1answer
305 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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308 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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1answer
582 views

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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274 views

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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175 views

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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1answer
131 views

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
797 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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86 views

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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1answer
254 views

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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1answer
176 views

References for Hopf Galois module theory

I am a first-year 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 ...
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101 views

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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1answer
265 views

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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188 views

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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82 views

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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