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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Galois elements determined by action on $n$-th roots of rationals?
Can there be an element $\sigma$ of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$, other than the identity and complex conjugation, which is completely determined up to conjugacy by its action on $\sqrt[n]r$...
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About some kind of "converse" of a theorem from Galois theory
It is a well known result from Galois theory that, given a Galois extension $L$ of a field $K$, an element $x$ of $L$ is in $K$ if for all $\sigma$ in $Gal(L/K)$, one has $\sigma(x)=x$.
My question ...
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Weyl theorem for non specified primitive root of unity
Let $\omega=e^{2i \pi/p}$.
Weyl theorems give all representations of matrix algebra span by $A,B$ such that either
$AB=\omega BA, A^p=B^p=I$,
or
$(k,l)\mapsto A^kB^l$ is a irreducible ...
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Monodromy groups from Galois's viewpoint
According to the Wikipedia article about monodromy, the monodromy group can be defined in terms of Galois theory in following way:
Let $F(x)$ denote the field of the rational functions in the ...
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Earliest use of the term "Galois extension"?
Does anyone know the earliest use of the term "Galois extension"? I thought it might be in Emil Artin's Notre Dame lectures but I couldn't find it there. (He does use the terms "normal" and "separable....
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Nature of polynomials of the form $x^n-a$ over finite fields
I state the following theorem from Serge Lang's Book- Algebra(3rd edition).
Theorem: Let $k$ be a field and $n$ an integer $\geq$ 2. Let $a\in k, a\neq 0$. Assume that for all primes $p$ such that $...
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Dimension of fixed vectors of a semi-linear operator
Let $L$ be a field with a field embedding $\sigma:L \rightarrow L$, and $K=L^{\sigma}$ be the fixed field of $\sigma$. For $A \in M_n(L)$ a matrix, consider the set $X=\{x \in L^n|Ax=\sigma(x) \}$ ...
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Dedekind's original proof of independence theorem
The independence theorem says, roughly, that a family of $\sigma\in\mathrm{Aut}_k(K)$ is $K$-linearly independent. In his Algebra, Serge Lang gave a proof of independence theorem following Artin (...
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Galois descent for dimension of vector spaces
Let $L/K$ be a Galois extension (I am interested in $\overline{\mathbb{Q}}/\mathbb{Q}$ so I do not assume it to be finite).
Let $V\subset L^n$ be a $L$-subvector space, of dimension $d$, such that $g(...
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Classification of mod p Galois Representations for l not equal to p
Let $l\neq p$ be primes and let $\text{G}_l:=\text{G}_{\mathbb{Q}_l}$. Let $k$ be a finite field of characteristic $p$ and $\bar{\rho}:\text{G}_l\rightarrow \text{GL}_2(k)$ a local Galois ...
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Analogy between metric space completion and algebraic closure
I've noticed some similarities between the story of completing a metric space and taking algebraic closure of a field. My question is whether these two stories can be generalized.
Metric space
Fix a ...
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A summation of powers defined by an equation over finite fields
Let $p$ be an odd prime and let $k\mid q$ for some positive integer numbers $k$ and $q$. Suppose that $r \in \mathbb{F}_{p^q}$ has multiplicative order $p^k-1$.
For each $1\leq u \leq p^k-3$, the ...
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Number Rings and (Galois) Descent
In algebraic number theory, one chooses for each finite étale $\mathbb{Q}$-algebra $K$ a finite $\mathbb{Z}$-algebra $\mathcal{O}_K$. Usually one simply speaks of the finite $\mathbb{Q}$-algebras ...
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Connection between Fourier analysis and Galois theory
Let $x\ \%\ m$ be the residue of $x$ modulo $m$, i.e.
$$x \equiv x\ \%\ m\pmod{m}$$
Let $\mu^n_m(x)$ denote multiplication by $n$ modulo $m$, i.e.
$$\mu^n_m(x) = nx\ \%\ m$$
Consider the Fourier ...
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Galois descent for profinite groups acting on local fields
Suppose that $G$ is a profinite group acting faithfully on a field $L$ and assume moreover that the action is admissible, that is, that every $l \in L$ is stabilized by an open subgroup of $G$. If ...
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Questions concerning the Fourier analysis of $ nx\ \%\ m$
Let $x\ \%\ m$ be the residue of $x$ modulo $m$, i.e.
$$x \equiv x\ \%\ m\pmod{m}$$
The plots of the functions $f_{nm}(x) = nx\ \%\ m$ exhibit characteristic patterns, especially periods of length $...
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Is this Related to Tannakian Formalism?
I am wondering how I might be able to express the following phenomenon, which is essentially equivalent to Artin's linear independence of characters, in Tannakian formalism. Any help would be much ...
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Elementary proof for Hilbert's irreducibility theorem
I have tried to find a complete proof for Hilbert's irreducibility theorem, but everything I found was way beyond my level of understanding.
I am only interested in the simple case where the ...
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Field K and integer n such that the etale fundamental group of (GL_n)_K is the profinite completion of GL_n(K)?
Prove or disprove:
Claim: there exists a field $K$ and an integer $n$ such that $\pi_1^{et}((GL_n)_K)$ is isomorphic to the profinite completion of the abstract group $GL_n(K)$?
Note that then $G_K \...
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Local polynomials of Frobenius-semisimple Weil representations which are tensor products of an Artin representation and an unramified character
Let $K$ be a local field and $\rho: W_K \to \operatorname{GL}(V)$ be a Weil representation. The for any finite extension $F/K$, we define the local polynomial
$$
P(\rho|_F,T) = \det{(1 - \operatorname{...
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On a special type of subring of $\mathbb C[x_0,...,x_{q-1}]$
Let $p,q$ be odd primes. Consider the polynomial ring $\mathbb C[x_0,...,x_{q-1}]$. For $m=0,1,...,p-1$, let
$$\sigma_m=\sum_{0\le j_0\le p;...;0\le j_{q-1}\le p; j_1+...+j_{q-1}=p; 1.j_1+...+(q-1)...
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Is there an alternative formula for solving cubic equations?
It is known that Cardano's formula for solving cubic equations is not good in the case of positive discriminants. In this case it expresses the solution through cubic roots of complex numbers. ...
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Is the following variant of Shafarevich's theorem known?
Let $Q$ be a finite simple group which may be realized as the Galois group of some extension of $\mathbb{Q}$ (like for instance $PSL_2(\mathbb{F}_p)$ for $p\geq 5$, or the monster group) and let $G$ ...
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Frobenius eigenvalues algebraic numbers
Let $X$ be a smooth projective variety over $\mathbf{F}_q$ and $\overline{X}$ its base change to $\overline{\mathbf{F}_q}$.
By Deligne’s Weil I, the eigenvalues of the geometric Frobenius acting on $...
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Unramified extensions of quadratic fields
Let $K/\mathbb{Q}$ be quadratic and let $L/K$ be an (everywhere) unramified Galois extension. If $L/K$ is abelian, then one can show that $L/\mathbb{Q}$ is Galois (eg see here). Is $L/\mathbb{Q}$ ...
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Is $\text{PSL}_2(\mathbb{F}_{p^m})$ known to be a Galois group over $\mathbb{Q}$ for $m>1$?
Let $\mathbb{F}$ be a finite field of characteristic $p$, is it known that $\text{PSL}_2(\mathbb{F})$ can be realized as a Galois extension of $\mathbb{Q}$ for any/all cases when $\mathbb{F}$ is not $\...
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Why Triangle of Mahonian numbers T(n,k) forms the rank of the vector space?
I am looking for an explanation of why Triangle of Mahonian numbers T(n,k) form the rank of the vector space $H^k(GL_n/B)$?
With respect to the property of Kendall-Mann numbers where the statement ...
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Does there exist an unramified $PSL_2(\mathbb{F}_p)$ extension of a quadratic field $K$?
Does there exist an unramified $PSL_2(\mathbb{F}_p)$ extension of a quadratic field $K$ for $p\geq 5$?
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Local factors determine Weil representations - proof of the Artin representation case
This post can be seen as a continuation of this post I created on MathOverflow.
I want to understand the proof of the following Theorem from "Euler Factors determine Weil Representations" by Tim and ...
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Elliptic curves and $GL(2)$ Iwasawa theory
Let $E$ be a elliptic curve without complex multiplication over a number field $F$. Let $F_n=F[E_{p^n}]$ and $F_{\infty}=F[E_{p^\infty}]$. So by a well know theorem of Serre, the Galois group $Gal(F_\...
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Correspondence Between First Galois Cohomology and Semilinear Actions Up to Isomorphism
I've been stuck for a while on Exercise 1.9 of Bjorn Poonen's "Rational Points on Varieties". We start with $L/K$ a finite Galois extension with Galois group $G$, some $r \in \mathbb{Z}_{\geq 0}$, and ...
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Local factors determine Weil representations - proof of the cyclic case
I already created this post on Math Stack Exchange but I was not so sure if this question fits better here. If it is not, I want to apologize in advance and feel free to delete my post.
I want to ...
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Reference request: Volume 2 of Abhyankar's lectures on algebra?
Abhyankar has a magnificent, if meandering (check them out if you want to see what I mean), set of lectures on algebra.
The description:
This book is a timely survey of much of the algebra developed ...
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Classes of curves with "determinant-like operation"
Consider a motivating example:
Let $E\in \mathbb{Q}[y][x]$ be of degree $n=2$ (in $x$) and separable when viewed as a member of $\mathbb{Q}[x,y]$. Therefore we can calculate it's roots in $\mathbb{Q}[...
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Relationships among constructions of fundamental group for schemes
There seem to be several constructions of fundamental group for schemes and stacks: by Grothendieck, Deligne, Nori, Noohi, Esnault-Hai, Vakil-Wickelgren, perhaps others as well. I am trying to ...
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Interpolation of families of local fields
Let $\Sigma$ be the set of all places of $\mathbf{Q}$. We’ll call, for any $v\in\Sigma$, a local field complete with respect to an absolute value lying over $v$ and of finite degree over $\mathbf{Q}_v$...
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“Algebraization" of $p$-adic fields
Part 1: a single finite place. Let $K$ be a finite extension of $\mathbf{Q}_p$.
Does there exist a number field $F/\mathbf{Q}$ and a finite place $v$ lying over $p$, such that for the completion of $...
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Having a separable extension of degree $n$ implies having a Galois extension of degree $n$?
I would like an explanation for the fact stated in the title. To repeat:
Question: How does one prove that if a field has a separable extension of degree $n$, then it has a Galois extension of ...
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Cyclic cubic extensions and Kummer theory
The Galois cohomology group $H^1(\mathbb{Q}, \mathbb{Z}/3\mathbb{Z})$ classifies cyclic cubic extensions $K/\mathbb{Q}$ (specifically: the non-trivial elements correspond to Galois cubic field ...
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Image of the norm map for degree $3$ galois extension over $\mathbb{Q}$
I want to construct a cyclic division algebra of degree $3$ over some degree $3$ Galois extension $E$ of $\mathbb{Q}$. So the construction is as follows:
As a set $D=E\oplus uE \oplus u^2 E$ where $u$...
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Are all real-closed subfields of $\overline{\mathbb{Q}}$ conjugate?
Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. The absolute galois group $G_\mathbb{Q}$ of $\mathbb{Q}$ acts on the set of real-closed subfields of $\overline{\mathbb{Q}}$.
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Does there exist a number field, unramified over a predetermined finite set of primes of Q, such that the inverse regular Galois problem is correct for that number field?
The question is: for any finite group, $G$, and any finite set of primes (of $\mathbb{Z}$), $P$, is there a number field $K$, such that there is a regular $G$-Galois extension of $\mathbb{P}^1_K$, and ...
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Recognizing the Galois group from the field discriminant
Along the lines of the general question "How much does the discriminant of a number field reveal about the field?", I was wondering how often it happens that the discriminant of some number field ...
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Shafarevich's theorem about solvable groups as Galois groups
I am seeking references to any proofs of Shafarevich's theorem about solvable groups being Galois groups.
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The inverse Galois problem, what is it good for?
Several years ago I attended a colloquium talk of an expert in Galois theory. He motivated some of his work on its relation with the inverse Galois problem. During the talk, a guy from the audience ...
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Normal closure and separable elements
Let $K\subset E\subset\bar{K}$ be field extensions, $\bar{K}$ an algebraic closure of $K$. Denote $E_s$ the field of separable elements of $E$ over $K$, denote $\tilde{E}\subset\bar{K}$ the normal ...
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Unique factorisation of prime geodesics?
In T. Sunada's 1985 paper ``Riemannian coverings and isospectral manifolds'', it is noted that for a compact Riemannian manifold $M$, prime (closed and non self-intersecting) geodesics behave like ...
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Does the $G$-norm coincide with the ordinary norm for "quasi-$G$-Galois" extensions
Let $S$ be a commutative ring, let $G$ be a finite group acting on $S$ via automorphisms (not necessarily faithfully), and
let $R$ be a subring of $S$ consisting of elements fixed $G$.
The extension $...
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Centraliser of an absolute Galois group
Let $K$ be a finite extension of $\mathbb{Q}_p$.
Is the centraliser of $\operatorname{Gal}(\overline{K}/K)$ in $\operatorname{Gal}(\overline{\mathbb{Q}_p} / \mathbb{Q}_p)$ trivial ?
If yes, how can I ...
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What are traces?
Let $A$ be a Noetherian commutative ring and Let $A\rightarrow B$ be a finite flat homomorphism of rings. We can thus form the so called "trace" $\mathrm{Tr_{B/A}}:B\rightarrow A$, which is a ...
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