Question

Prove that the Galois group of $f\left(x\right)$ over $\mathbb{Q}$ is not a simple group.

score 83 · Answer 1 · 2026-04-16 06:17:17

83

Views

Prove that the Galois group of $f\left(x\right)$ over $\mathbb{Q}$ is not a simple group.

Published on 16 Apr 2026 - 6:17

score 841 · Answer 2 · 2026-04-16 03:47:14

841

Views

Is the extension $\mathbb{Q}(\sqrt{5},\sqrt{7})$ simple? If so/not, why?

Published on 16 Apr 2026 - 3:47

score 513 · Answer 3 · 2026-04-17 04:02:06

513

Views

Galois Group and Subfield lattice of $(x^2 - 7)(x^4 + x^3 + x^2 + x + 1)$ over $\mathbb{Q}$

Published on 17 Apr 2026 - 4:02

score 54 · Answer 4 · 2026-04-10 19:04:27

54

Views

Why a homomorphism from full Galois group of rationals cannot factor.

Published on 10 Apr 2026 - 19:04

score 1.5k · Answer 5 · 2026-05-05 10:03:00

1.5k

Views

How to compute a primitive element for the splitting field of $x^3-2 \in \Bbb{Q}[x]$?

Published on 05 May 2026 - 10:03

score 308 · Answer 6 · 2026-04-13 18:13:13

308

Views

Automorphism of $\mathbb{C}$ over a subfield $K$ of $\mathbb{C}$

Published on 13 Apr 2026 - 18:13

score 200 · Answer 7 · 2026-04-10 06:36:50

200

Views

The degree of extension

Published on 10 Apr 2026 - 6:36

score 266 · Answer 8 · 2026-04-16 19:44:11

266

Views

Let $K/E$ and $E/F$ be Galois extensions. If every $σ ∈ \text{Aut}(E/F)$ is the restriction of an element of $\text{Aut}(K/F)$, then $K/F$ is Galois.

Published on 16 Apr 2026 - 19:44

score 137 · Answer 9 · 2026-04-17 12:34:48

137

Views

Suppose $a,b\in E$ and $\sigma\in \textrm{Gal}(E/F)$ with $\sigma(a)=b$. Show that $a$ and $b$ have the same minimal polynomial in $F[x]$

Published on 17 Apr 2026 - 12:34

score 39 · Answer 10 · 2026-04-14 18:51:29

39

Views

If $K$ is a subfield of $\mathbb{C}$ then $K(\zeta)/K$ is Galois

Published on 14 Apr 2026 - 18:51

score 315 · Answer 11 · 2026-05-11 08:29:39

315

Views

Efficiently calculating the Galois group of $p(x)=x^4+4x^2-2$

Published on 11 May 2026 - 8:29

score 66 · Answer 12 · 2026-04-16 07:53:16

Let $\alpha=2^{1/5}$ ,$\zeta=e^{2i\pi/5}$ and $K=\mathbb{Q}(\alpha\zeta)$, can the field automorphism of K be extended to an automorphism of C?

score 339 · Answer 13 · 2026-04-14 13:11:41

339

Views

My attempt at finding the Galois group of $E:\Bbb Q$, where $E$ is the splitting field of $x^3-5$

Published on 14 Apr 2026 - 13:11

score 191 · Answer 14 · 2026-05-11 09:32:53

191

Views

Group is soluble if and only if quotient is abelian

Published on 11 May 2026 - 9:32

score 367 · Answer 15 · 2026-05-10 18:03:58

367

Views

Find all subfields of $\mathbb{Q}(\mu_{24})$

Published on 10 May 2026 - 18:03

List Question

Prove that the Galois group of $f\left(x\right)$ over $\mathbb{Q}$ is not a simple group.

Is the extension $\mathbb{Q}(\sqrt{5},\sqrt{7})$ simple? If so/not, why?

Galois Group and Subfield lattice of $(x^2 - 7)(x^4 + x^3 + x^2 + x + 1)$ over $\mathbb{Q}$

Why a homomorphism from full Galois group of rationals cannot factor.

How to compute a primitive element for the splitting field of $x^3-2 \in \Bbb{Q}[x]$?

Automorphism of $\mathbb{C}$ over a subfield $K$ of $\mathbb{C}$

The degree of extension

Let $K/E$ and $E/F$ be Galois extensions. If every $σ ∈ \text{Aut}(E/F)$ is the restriction of an element of $\text{Aut}(K/F)$, then $K/F$ is Galois.

Suppose $a,b\in E$ and $\sigma\in \textrm{Gal}(E/F)$ with $\sigma(a)=b$. Show that $a$ and $b$ have the same minimal polynomial in $F[x]$

If $K$ is a subfield of $\mathbb{C}$ then $K(\zeta)/K$ is Galois

Efficiently calculating the Galois group of $p(x)=x^4+4x^2-2$

Let $\alpha=2^{1/5}$ ,$\zeta=e^{2i\pi/5}$ and $K=\mathbb{Q}(\alpha\zeta)$, can the field automorphism of K be extended to an automorphism of C?

My attempt at finding the Galois group of $E:\Bbb Q$, where $E$ is the splitting field of $x^3-5$

Group is soluble if and only if quotient is abelian

Find all subfields of $\mathbb{Q}(\mu_{24})$

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