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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A mixed Vandermonde-Wronskian matrix
I am trying to prove that a matrix of the following form is generically nonsingular:
$A= \begin{bmatrix}
1&1&1&1 \\
f_1 & f_2 & f_3 &f_4 \\
(f_1 -\frac{d}{dt}).f_1& (...
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Infinite simple Galois groups
Conjecturally, every finite group is the Galois group of some extension of the rationals.
This question made me wonder what is known about infinite
simple groups occurring as Galois groups.
What ...
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Galois descent for absolute Galois group
Let $K$ be a field of characteristic zero, $\bar{K}$ its algebraic closure and $X$ a smooth, projective $K$-scheme. We know the Galois descent theory for quasi-coherent sheaves defined on $X_L$ for a ...
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Is $699 \ldots 998$ value of the Euler totient function?
Let $n_l = 7 \cdot 10^{l+1}-2 = 699\ldots 998$. I want to know if there is an $l$ such that $n_l$ can occur as the value of the Euler totient function $\varphi(n)$. For given $l$, it is easy to check ...
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Why are procyclic subgroups of Galois groups of number fields free profinite?
On p832 of Coombes, Harbater - Hurwitz familes and arithmetic Galois groups, the following is claimed:
Let $K$ be a number field, take $1 \neq \omega \in \mathrm{Gal}(\bar{\mathbb{Q}}/K)$, and let $...
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Galois extensions inside a division ring
Let $D$ be a division ring which has finite dimension over its centre.
Q1. Under which conditions can one find a maximal subfield $K$ of $D$ and a proper subfield $L$ of $K$ such that $K/L$ is Galois?...
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Why does the Galois twist of this cover specialize to a certain field extension?
I didn't feel MO was the best place to ask this question, so apologies for this, but when I asked it at https://math.stackexchange.com/questions/2297837/why-is-this-cubic-polynomial-generic-for-cyclic-...
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Relation between Galois theory and Etale Cohomology
I am a graduate student working on category theory. I am familiar with categorical Galois theory, in the way developed by Janelidze - as described for example in "Galois Theories". I am now trying to ...
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Grothendieck's Galois Theory today
I have recently become aware of, and started to study in my free time (abundant in these summer months) Grothendieck's Galois Theory (GGT), as formulated in SGA 1 and later by Grothendieck's ...
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An Explicit Example of Galois Theory for Schemes
I'm currently attempting to understand Galois theory for schemes, largely following the books Galois Theory for Schemes by Henrik Lenstra and Galois Groups and Fundamental Groups by Tamas Szamuely. ...
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Minimal polynomial of degree 6
Let $F$ be a field of characteristic not equal to $2$.
Assume $L/F$ is a field extension of degree $6$ with no intermediate subfields.
We know by a result due to Joubert (1867), later improved by H. ...
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When is a Fourier coefficient field Galois?
Let $f$ be a modular form with $q$-coefficients $a_n$, and let $L=\mathbf Q(a_n:n\ge 0)$ be the Fourier coefficient field.
Does anyone know of any necessary or sufficient conditions for $L/\mathbf Q$...
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"Understanding" $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$
I have heard people say that a major goal of number theory is to understand the absolute Galois group of the rational numbers $G = \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. What do people mean when ...
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Degree required to generate a particular galois group
Let $L/\mathbb{Q}$ be a Galois closed field with Galois group a subgroup of $S_6$. Is it the case that $L$ is the compositum of Galois closures of linearly disjoint fields of degree at most $6$ over $\...
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An algebraic number is not a root of unity?
This problem is related to my study of the Burau representation of the braid group $B_3$: I was trying to show that certain "congruence subgroups" are of infinite index.
There is an approach that ...
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Is there an algebraic formula for the eigenvalues of a symmetric $n\times n$ matrix?
Is there a formula in radicals for the eigenvalues, or at least the largest eigenvalue, of an $n\times n$ symmetric matrix, in terms of the entries? For what $n$ is there a formula? There obviously ...
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Fundamental groups of topoi
Just yesterday I heard of the notion of a fundamental group of a topos, so I looked it up on the nLab, where the following nice definition is given:
If $T$ is a Grothendieck topos arising as category ...
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Inverse galois problem and étale homotopy
Is there any relation between étale homotopy theory (Grothendieck-Galois theory) and the inverse Galois problem?...I mean...in classical homotopy theory, every finite group $G$ realizes as a "Galois ...
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What are Galois Categories used for?
Galois categories are introduced (for the first time?) in SGA1, but here's an English introduction that's available online: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Lynn.pdf
It ...
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Noether-Deuring for injections and surjections?
Noether-Deuring theorem (not in the strongest form, but in the one I usually need):
Let $L\diagup K$ be a field extension. Let $A$ be a $K$-algebra which is finite-dimensional as a vector space over $...
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Are there analytical solutions for the polynomial $\alpha K x^\alpha + x -N = 0$?
I'm interesting in finding analytical solutions for the equation
$$\alpha K x^\alpha + x -N = 0,$$
where $\alpha$ is a positive integer and both $K$ and $N$ are positive real constants.
Based on ...
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Existence of "splitting objects" and algebraic closures
Definition 0. Let $(\mathbf{C},U)$ denote a concrete category. Let "splits" be a desirable property that pointed $\mathbf{C}$-objects may or may not satisfy. Let $(X,x)$ be a pointed $\mathbf{C}$-...
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complex numbers over algebraic numbers, continuous cohomology
This question is related to this one. Choose an embedding $\overline{\mathbf{Q}}\rightarrow \mathbf{C}$ from the algebraic closure of the field of rational numbers to the field of complex numbers. Is ...
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Cyclotomic Extension of a Perfectoid Space
Maybe, I am being stupid, but when I consider ramified extension of a perfectoid field with the characteristic $0$, I cannot find the correspondent field with characteristic $p$. Let me put it more ...
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Splitting of polynomials over rational function fields
Let $K$ be a number field, and let $P(t,X)$ be a monic polynomial in $X$ with coefficients in $K(t)$.
I would like to understand the set $T$ consisting of those $t_0 \in K$ such that the polynomial $...
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Is co-restriction in Galois cohomology in fact the norm map via Kummer isomorphism?
Let $\mathrm{F}$ be a field that contains a root of unity of order $p$, where $p$ is a prime number. Fix an element $a$ such that $a \in \mathrm{F}$ and $\sqrt[p]{a} \notin \mathrm{F}$. Consider the ...
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Inverse Galois problem on the upper or lower numbering filtration
Let $K$ be a (complete) discrete valuated field and $E$ a Galois extension of group $G$. Then one has two filtrations on $G$, the upper and the lower numbering.
Assume that $K$ is equal to its ...
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Galois theory for products of fields (aka finite etale extensions)
Let $F$ be a field. By a Galois algebra over $F$ I mean a finite etale extension, that is, a product $K = K_1 \times \cdots \times K_r$ of finite (separable) field extensions, of total degree $[K : F]...
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Is it possible to solve $P = Cny^{-1}(1-1/(1+y/n)^{nT}) + M/(1+y/n)^{nT}$ for $y$? [closed]
The equation
$$
P = \frac{Cn}{y}\left(1-\frac{1}{(1+\frac{y}{n})^{nT}}\right)+\frac{M}{(1+\frac{y}{n})^{nT}}
$$
represents the present value (price $P$) of a government bond which pays $C$ ...
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Correspondence between coverings and field extensions
I am self reading from Groups as Galois Group by Helmut Volklein
There is a result on page 94(section 5.4)
Let $G$ be a finite group. Let $P\subset P^{1}$ finite and $q\in P^{1}\P$. There is a ...
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Galois action on units of totally real cyclic numberfields
Given a cyclic totally real number field $K/\mathbb{Q}$ of degree n with unit group isomorphic to $\mathbb{Z}/2 \times \mathbb{Z}^{n-1}$, how much is known about the action of Gal$(K/\mathbb{Q})$ on ...
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Totally ramified subextension in a finite extension of $\mathbf{Q}_p$
Let $K$ be a finite extension of $\mathbf{Q}_p$. Let $F_d$ be the unramified extension of $\mathbf{Q}_p$ of degree $d$. I would like to know whether there exists some $d \geq 1$ and some $L \subset K \...
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Is $n^{n-1}+(-1)^{n+1}(n-1)^{n-1}$ a squarefree number?
I am trying to prove that $f(x)=x^n+nx+n$ has Galois group $S_n$ over the rationals.
The discriminant of this polynomial is $\Delta= (-1)^{n(n-1)/2}n^n(-(n-1)^{n-1}+(-1)^n n^{n-1})$. The Newton ...
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Forcing as a new chapter of Galois Theory?
There is a (very) long essay by Grothendieck with the ominous title La Longue Marche à travers la théorie de Galois (The Long March through Galois Theory). As usual, Grothendieck knew what he was ...
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Galois group and invariant polynomials
Let $K$ be a field and $\alpha_i$ ($i=1,\dots,n$) be Galois conjugates. Let $L=K(\alpha_1,\dots,\alpha_n)$ and $G=Gal(L/K)$. We embed $G$ in $S_n$ by its action on $\alpha_i$. Let $H$ be another ...
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Neat applications of Galois descent?
I'm enjoying reading about Janelidze's categorical Galois theory, which gives as a special case the usual theorems of Galois descent (along torsors). The approach I took was just with covering space ...
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Characterization of torsors which are locally trivial in terms of descent
Let $\mathsf C$ be a category and $G$ an internal group. Suppose $\mathsf C$ is finitely complete, so that $\pi_2:G\times B\to B$ is an internal group in $\mathsf C_{/B}$ for every $B$. A $G$-bundle ...
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Geometry of componentially locally strongly separable algebras
Janelidze's categorical Galois theory yields, for nice adjunctions, a good notion of covering morphism.
The category of finitely affine schemes admits such an adjunction into the category of ...
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Galois group of an L-function
Let $ M $ be a class of L-functions such that whenever $ F $ and $ G $ belong to $ M $, then so do their product $ F.G $ and their tensor product $ F\otimes G $ defined by $ F\otimes G : s\...
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Why should the anabelian geometry conjectures be true?
I had probed friends of mine about Grothendieck's motivation for making the anabelian geometry conjectures, and they gave me the following explanation:
If $X$ is a hyperbolic curve over some field $K$...
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Anabelian geometry study materials?
I want to study anabelian geometry, but unfortunately I'm having difficulties in finding some materials about it. If you could offer me some books/papers/articles I would be glad.
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What "should" be the absolute galois group of a field with one element
As far as I know there is many "suggestions" of what should be a "field with one element" $\mathbf{F}_{1}$.
My question is the following:
How we should think or what should be the "absolute Galois ...
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Parametric Solvable Septics?
Known parametric solvable septics are,
$$x^7+7ax^5+14a^2x^3+7a^3x+b=0\tag{1}$$
$$x^7 + 21x^5 + 35x^3 + 7x + a(7x^6 + 35x^4 + 21x^2 + 1)=0\tag{2}$$
$$x^7 - 2x^6 + x^5 - x^4 - 5x^2 - 6x - 4 + n(x - ...
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The holomorphic version of Galois theory
We identify the space of polynomials of degree n with $\mathbb{C}^{n+1}-\mathbb{C}^{n}$, that is an $n+1$ tuple $(a_{n},a_{n-1},\ldots,a_{0})$ with $a_{n} \neq 0$ is identified with $p(z)=a_{n}z^{n} +...
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Consequences of Shafarevich conjecture
The Shafarevich conjecture states that the Galois group $\mathrm{Gal}({\overline{\mathbf{Q}}/\mathbf{Q}^{ab}})$ is a free profinite group, where $\mathbf{Q}^{ab}$ is the maximal abelian extension of $\...
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Weyl Groups as Galois groups
I am looking for explicit examples (for all positive integers $n \ge 5$) of degree $2n$ even polynomials $f(x)=h(x^2)$ over the field $\mathbb{Q}$ of rational numbers such that the Galois groups of $...
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small Pisot numbers with real conjugates
I have read a wikipedia article on Pisot number or PV number: https://en.wikipedia.org/wiki/Pisot%E2%80%93Vijayaraghavan_number
I define a Pisot number $\alpha$ is "small" iff $\alpha<2$.(It is ...
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Five cubes, Hadamard and Shklyarskiy
Here is my(=bad) translation of from the paper about Shklyarskiy by Golovina:
... in 1937/38 Dodik presented to school students a complete proof of Abel's theorem about equations of degree 5. He ...
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Resolvent in French [closed]
First, I apologize if the question doesn't fit this forum.
In a thread about Galois theory on a French math forum, I read "le sextique résolvent" and the spelling looks odd to me. I would have ...
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On the partner of the Emma Lehmer quintic
Given,
$$x^5+10cx^3+10dx^2+5ex+f = 0$$
If there is an ordering of its roots such that,
$$\small x_1 x_2 + x_2 x_3 + x_3 x_4 + x_4 x_5 + x_5 x_1 - (x_1 x_3 + x_3 x_5 + x_5 x_2 + x_2 x_4 + x_4 x_1) = 0\...
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