Questions tagged [galois-extensions]

Question 1

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 2

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 3

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 4

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 5

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 6

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 7

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 8

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 9

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 10

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 11

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 12

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 13

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 ...

Question 14

Suppose $G$ is a finite group, $F$ is a field of characteristic zero, and $L/F$ is a field extension. Let $\varphi$ be an irreducible representation of $G$ over $F$, with character $\chi$, and in $L$ ...

Question 15

While the goal is multiplying polynomials, we can immediately reduce the problem to evaluating polynomials, then to multiply the evaluations. If one has a polynomial with coefficients and basis ...

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Questions tagged [galois-extensions]

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?

Dimension of decomposition of an irreducible representation on a non-algebraically closed field

Fast multiplication of polynomials with GF(2) evaluations

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