# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $[\mathbb{Q}_1:\mathbb{... 1answer 176 views ### References for Hopf Galois module theory I am a first-year PhD student and I am really interested in Galois module theory, both in a "classical" and in a "nonclassical" sense. In the last months I have been reading about ... 0answers 101 views ### Compass and straightedge construction of Poncelet polygons Gauss–Wantzel theorem states that A regular n-gon is constructible with straightedge and compass if and only if$n = 2^kp_1p_2...p_t$, where$p_i$'s are distinct Fermat primes (A Fermat prime is a ... 1answer 265 views ### Origin and context of adjunctions inducing equivalences between full subcategories The following is well-known. Theorem. Let$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)))_{...
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 ...