# 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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### Is $E$ finite and normal? [closed]

Let $E$ be a field and $H$ a finite subgroup of the group Aut $E$ of all automorphisms of $E.$ Let $F=$ Inv $H$ be the field of invariants of $H$. Is $E$ finite and normal over $F$? Is every element ...
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### Galois groups of special polynomials

This question is motivated by long experiments with GAP. Call a monic polynomial with integer coefficients special in case it is irreducible and has only coefficients $-1$, $0$ or $1$. Let $n \geq 5$....
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### Galois group of polynomials related to Fibonacci and Catalan numbers

Let $F_n$ be the Fibonacci and $C_n$ the Catalan numbers. Define a polynomial by $F_n(x):=\sum\limits_{k=1}^{n}{F_k x^{n-k}}$. For example $F_8(x)=x^7+x^6+2x^5+3x^4+5x^3+8x^2+13x+21$. And another ...
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### Cubic polynomials over finite fields whose roots are quadratic residues or non-residues

For a cubic polynomial $f(x)=x^3+x^2+\frac{1}{4}x+c$ over $\mathbb{F}_q$, where $q$ is a odd prime power, I find that for a lot of $q$, there does not exist $c\in\mathbb{F}_q$ such that $f$ has three ...
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### Any way around Abel's impossibility theorem?

Abel's impossibility theorem states that the roots of a general polynomial (of degree 5 or higher) cannot be written using arithmetic operations and radicals. Radicals are solutions of a specific ...
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### Conjugation action of $Gal(\bar{s}/s)$ on the tame ramification group

There is a statement in SGA 7-1 Exposé 1 (P. Deligne, Résumé des premiers exposés de A. Grothendieck, pdf of SGA7-1), (0.3.1): $S$ is a Henselian trait (i.e. the spectrum of a henselian discrete ...
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### Algorithm to compute minimal polynomials

Suppose $L/K$ is a finite Galois extension of fields of degree $n$. Suppose that we know an irreducible polynomial $f\in K[x]$ such that $L\cong K[x]/(f)$. Suppose also that we know the Galois group ...
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### Non-linear Galois descent

This question is about Galois theory. So let $K / k$ be a Galois extension of fields. Let us assume that $K / k$ is finite dimensional, though everything can be made to work in the profinite case by ...
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### Lang's proof of the Shimura's exact sequence

Let $E$ be an elliptic curve having invariant $j$ and defined over $\mathbf C(j)$. Let $\sigma$ be an automorphism of $F_{\mathbf C}$, the field of all modular functions of all levels. Let $p$ be a ...
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### Explicit construction of abelian wild inertial extensions of maximal tamely ramified extension of $\mathbb{Q}_p$?

In Iwasawa's paper On Galois groups of local fields, he proves that if $V$ is the maximal tamely ramified extension of $\mathbb{Q}_p$, with Galois group $\Gamma$ over the base, then its abelianized ...
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### split integral model of a reductive group

Let $F$ be a number field, $p\in\mathbb{Z}$ a prime which is unramified in $F$ and $G$ a connected reductive group over $F$. Moreover $G$ is supposed to be quasi-split over $p$. Does there exist a ...
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### Splitting field of an intermediate field

Consider the following 'wrong' question. Let $f(x) \in F[x]$ be an irreducible polynomial in a polynomial ring of a field $F$. Let $L$ be the splitting field of $f(x)$ over $F$. Assume that $L$ is a ...
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### Fontaine - Wintenberger field of norms and imperfect case

Let $K$ be a complete discrete valued field whose residue field $k_K$ has characteristic $p$ and has the property that $[k_K:k_K^p]=p^d$ for some $d$. Let $t_{\alpha}, 1 \leq \alpha \leq d$ be a set ...
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### Local Class field theory for curves over $p$-adic fields

Let $X$ be a smooth curve over $\operatorname{Spec}\mathbb{Q}_p$ and $P\in X(\mathbb{Q}_p)$. Let $K_P\simeq \mathbb{Q}_p((T))$ denote the completion of the function field of $X$ at $P$. What is known ...
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### p-adic field extension of degree 2n without a subfield of degree 2?

I need an example of a p-adic field extention $L/F$ of degree $[L:F]=2n$ without a subfield $K\subset L$ of degree $[K:F] = 2$.
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### Collinear Galois conjugates

This is inspired by this old question, which may provide a bit more background. But the two present questions seem somewhat more fundamental to me. Let $p$ be an irreducible polynomial with integer ...
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### Non-cyclic Galois groups over the field of formal Laurent series in positive characteristic

This should be an easy question, but I am unfortunately not able to give an answer, so I am sorry if this is not the appropriate level for the site. Let $C$ be an algebraically closed field of ...
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### Abel-Ruffini theorem for systems of polynomial equations

I know the Abel-Ruffini theorem, which states that a general polynomial equation in one variable with degree $\geq 5$ is not solvable in radicals. I wonder whether there is a similar result for ...
From Galois theory, we know that $[\mathbb{Q}(\sqrt{2}, \sqrt{3}, \dots \sqrt{p_k}) : \mathbb{Q}] = 2^k$. Suppose I plug in rational approximations to the square roots, then of course the classical ...