Question

Galois action on $K\otimes_{\mathbb{Q}} \mathbb{Q}_\ell$

score 75 · Answer 1 · 2026-03-27 00:58:10

75

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Galois action on $K\otimes_{\mathbb{Q}} \mathbb{Q}_\ell$

Published on 27 Mar 2026 - 0:58

score 633 · Answer 2 · 2020-11-29 12:04:58

633

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Let $f(x)= x^3+ax^2+bx+c \in \mathbb{Q}[x]$. Show that the splitting field of $f$ over $\mathbb{Q}$ has degree 1, 2, 3 or 6 over $\mathbb{Q}$.

Published on 29 Nov 2020 - 12:04

score 508 · Answer 3 · 2026-03-28 06:59:56

508

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Prove that the Galois group $G = Gal(x^5-2; \mathbb{Q})$ is a group metacyclic.

Published on 28 Mar 2026 - 6:59

score 272 · Answer 4 · 2020-12-02 17:56:19

272

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Non cyclic Galois extension of degree 12

Published on 02 Dec 2020 - 17:56

score 150 · Answer 5 · 2020-12-08 04:24:46

150

Views

Norm of a polynomial $N_{E/K}(f)(X)$ with $E/K$ a Galois extension

Published on 08 Dec 2020 - 4:24

score 36 · Answer 6 · 2020-12-13 12:59:01

36

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Prove that $\operatorname{Gal}(K_1 / F_1)$ and $\operatorname{Gal}(K_2 / F_2) $ are isomorphic

Published on 13 Dec 2020 - 12:59

score 42 · Answer 7 · 2020-12-13 14:57:06

42

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$E=K(a_1,...,a_n)$ and $\sigma \in Gal(E/K)$ such that $\sigma (a_i)=a_i$ for all $i$. Prove $\sigma = 1_E$

Published on 13 Dec 2020 - 14:57

score 65 · Answer 8 · 2020-12-15 12:46:30

65

Views

Is the extension $ℚ(\alpha):ℚ(\alpha^2)$ a Galois extension?

Published on 15 Dec 2020 - 12:46

score 67 · Answer 9 · 2020-12-15 21:10:08

67

Views

Is $(x-t)^p$ irreducible?

Published on 15 Dec 2020 - 21:10

score 118 · Answer 10 · 2020-12-16 04:42:39

118

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Splitting field of an infinite family of separable polynomials is infinite dimensional if there is no finite subfamily of which it is splitting.

Published on 16 Dec 2020 - 4:42

score 31 · Answer 11 · 2020-12-17 07:42:43

31

Views

Galois group is dihedral iff $\exists \beta_i\in F$

Published on 17 Dec 2020 - 7:42

score 199 · Answer 12 · 2020-12-17 20:18:56

199

Views

For a recursion based on $x+y+z+2\sqrt{xy+yz+zx},$ does what happens in $\mathbb Z\left[\sqrt n\right]$ stay in $\mathbb Z\left[\sqrt n\right]$?

Published on 17 Dec 2020 - 20:18

score 67 · Answer 13 · 2020-12-24 18:15:07

67

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Find the degree of the normal closure $E$ of $F$ over $\mathbb Q$. What is the degree of $E$ over $\mathbb Q$?

Published on 24 Dec 2020 - 18:15

score 527 · Answer 14 · 2026-02-23 17:27:35

527

Views

Solvability by radicals of a separable field of characteristic $p$

Published on 23 Feb 2026 - 17:27

score 148 · Answer 15 · 2020-12-28 01:53:54

148

Views

Possible degrees of an algebraic element in intermediate fields

Published on 28 Dec 2020 - 1:53

List Question

Galois action on $K\otimes_{\mathbb{Q}} \mathbb{Q}_\ell$

Let $f(x)= x^3+ax^2+bx+c \in \mathbb{Q}[x]$. Show that the splitting field of $f$ over $\mathbb{Q}$ has degree 1, 2, 3 or 6 over $\mathbb{Q}$.

Prove that the Galois group $G = Gal(x^5-2; \mathbb{Q})$ is a group metacyclic.

Non cyclic Galois extension of degree 12

Norm of a polynomial $N_{E/K}(f)(X)$ with $E/K$ a Galois extension

Prove that $\operatorname{Gal}(K_1 / F_1)$ and $\operatorname{Gal}(K_2 / F_2) $ are isomorphic

$E=K(a_1,...,a_n)$ and $\sigma \in Gal(E/K)$ such that $\sigma (a_i)=a_i$ for all $i$. Prove $\sigma = 1_E$

Is the extension $ℚ(\alpha):ℚ(\alpha^2)$ a Galois extension?

Is $(x-t)^p$ irreducible?

Splitting field of an infinite family of separable polynomials is infinite dimensional if there is no finite subfamily of which it is splitting.

Galois group is dihedral iff $\exists \beta_i\in F$

For a recursion based on $x+y+z+2\sqrt{xy+yz+zx},$ does what happens in $\mathbb Z\left[\sqrt n\right]$ stay in $\mathbb Z\left[\sqrt n\right]$?

Find the degree of the normal closure $E$ of $F$ over $\mathbb Q$. What is the degree of $E$ over $\mathbb Q$?

Solvability by radicals of a separable field of characteristic $p$

Possible degrees of an algebraic element in intermediate fields

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