Questions tagged [galois-extensions]

Question 1

I'm trying to find the Galois group of $\mathbb{Q}\left(\zeta_{6}\right)/\mathbb{Q}$, where $\zeta_k=e^\frac{2\pi i}{k}$. We know that $\left\{1,\zeta_{6},\zeta_{6}^2,\zeta_{6}^3,\zeta_{6}^4,\zeta_{6}^...

Question 2

In Milne (Fields and Galois Theory), by definition constructible elements are in $\mathbb{R}$. However, it is also argued (in remark 3.27) that given an algebraic constructible number, its Galois ...

Question 3

It seems to me that the beginning of Galois theory (Galois's work) is still not clear to me. To prove the quintic or higher degree polynomial is not solvable in radicals, Evertise Galois considered ...

Question 4

How to find the degree and the ramification index of the splitting field of a certain polynomial over $\mathbb{Q}_p$? For instance, $f(x)=3+9x+3x^4+x^6$ over $\mathbb{Q}_3$? If $\gamma$ is one of the ...

Question 5

Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication by $\mathbb{Q}(i)$, e.g. any curve of the form $E:y^2 = x^3 + Ax$ with $A \in \mathbb{Q}^\times$. I would like to compute generators ...

Question 6

Let $K \subset F \subset E$ be a tower of extensions with $E/K$ being a Galois extension. How can I show that $g(F)=F \text{ for all }g\in {\rm Gal}(E/K) \iff F/K$ is normal? I tried using that $${\rm ...

Question 7

Let $L/K$ is a finite field extension and assume it a Galois extension. I want to prove the following simplest form of the Fundamental theorem of Galois theory: $(a)$ For any subgroup $H$ of $\...

Question 8

I am asking whether the following statement is true: given a prime $p$ a finite group $G$ of order coprime to $p$ an integer $g$ greater than 1 an algebraically closed field $K$ of characteristic $p$...

Question 9

I believe the following problem comes from the UCLA algebra qualifying exams. Let $K / F$ be a finite Galois extension of fields and $\alpha \in K \setminus F$. Let $F \subseteq E \subseteq K$ such ...

Question 10

I'm currently studying étale cohomology and its application to the Weil Conjectures, following the Lecture Notes of Milne. In Chapter 27, he defines and reviews the properties of the Frobenius ...

Question 11

Let $p$ be an odd prime, and let $n_1,\dots,n_m\in\mathbb F_p$ be the roots of $x^{\frac{p-1}{2}}-1\in\mathbb F_p[x]$. Does $\zeta_p^{n_1}+\cdots+\zeta_p^{n_m}$ generate $\mathbb Q(\sqrt{p})$ if $p\...

Question 12

Let $G$ be a finite group, $K$ be a field with characteristic zero, and $C$ be an algebraic closure of $K$. The algebras $CG$ and $KG$ are semisimple. Let $\chi$ be a $C$-character of $G$ ...

Question 13

I have a question about an answer to this post concerning finding fixed fields of subgroups of the Galois group $K$ of $f(x)=x^4-2$ over $\mathbb Q$. Part of the accepted answer involves determining ...

Question 14

I’m looking at the quotient ring $$ R := (\mathbb Z/2^{n}\mathbb Z)[X]\big/\bigl(X^{2^m}+1\bigr), $$ for example let's focus on $n=16,m=4$. I understand $X^{16}+1$ is irreducible over $\mathbb Z/2^{16}...

Question 15

Let $F$ be a field and $K$ be a finite extension of $F$. (Assume all these fields have characteristic zero). Take $\alpha\in K$. Let $n$ be the total number of conjugates of $\alpha$ over the base ...

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Questions tagged [galois-extensions]

Galois group of $\mathbb{Q}(\zeta_{6})$ over $\mathbb{Q}$ (reprise)

Constructibility of conjugates of constructible algebraic numbers

Which permutations Galois considered out of $n!$ permutations in $S_n$ to prove insolvability of quintic polynomial?

Splitting field of a polynomial over $\mathbb{Q}_p$?

Computing large division fields of elliptic curves with CM

Step in the proof of Galois fundamental theorem

How to prove the simplest statement of Galois correspondence theorem in Galois extension?

A question about coverings of the projective line in positive characteristic

Basic application of Galois correspondence

Computing degree of Frobenius endomorphism, following Milne's Étale Cohomology

If $n_1,\dots,n_m\in\mathbb F_p$ are the roots of $x^{\frac{p-1}{2}}-1$, does $\zeta_p^{n_1}+\cdots+\zeta_p^{n_m}$ generate $\mathbb Q(\sqrt{\pm p})$?

Character theory of finite groups and Schur group of fields with characteristic zero

How to rule out possible groups when calculating the Galois group of $x^4-2$ over $\mathbb Q$

Zero divisors in $(\mathbb Z/2^{16}\mathbb Z)[X]/(X^{16}+1)$

Does the number of conjugates which lie in a field extension divide the total number of conjugates?

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