# 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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### 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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### 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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### 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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### 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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### 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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### 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 ...