Question

Extensions of the quadratic closure of $\mathbb{Q}$

score 456 · Answer 1 · 2026-04-14 17:59:02

456

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Extensions of the quadratic closure of $\mathbb{Q}$

Published on 14 Apr 2026 - 17:59

score 465 · Answer 2 · 2026-04-11 03:41:23

465

Views

Degree of field extension over a fixed field

Published on 11 Apr 2026 - 3:41

score 610 · Answer 3 · 2026-04-17 11:01:29

610

Views

$K/\mathbb{Q}$ contains a complex number - is complex conjugation in $\text{Aut}(K/\mathbb{Q})$?

Published on 17 Apr 2026 - 11:01

score 347 · Answer 4 · 2026-04-15 14:38:55

347

Views

Example of a field extension $K/F$ such that $K$ is the splitting field of some (non-separable) polynomial in $F$, but not Galois over $F$?

Published on 15 Apr 2026 - 14:38

score 1.4k · Answer 5 · 2026-03-25 22:26:09

1.4k

Views

On the degree of cyclotomic fields extension.

Published on 25 Mar 2026 - 22:26

score 63 · Answer 6 · 2026-04-15 09:26:01

63

Views

For $a_{i}:=\sqrt{2+a_{i-1}}$, how to prove that $\mathbb{Q}\subset\mathbb{Q}(a_i)$ is cyclic of degree $2^i$?

Published on 15 Apr 2026 - 9:26

score 282 · Answer 7 · 2026-04-09 06:58:38

282

Views

Galois theory for beginners, mistake?

Published on 09 Apr 2026 - 6:58

score 908 · Answer 8 · 2026-04-14 22:24:12

908

Views

Show that a polynomial is still irreducible in a extension field

Published on 14 Apr 2026 - 22:24

score 74 · Answer 9 · 2026-04-13 03:45:52

74

Views

Degree of the extension field

Published on 13 Apr 2026 - 3:45

score 218 · Answer 10 · 2026-04-16 08:40:13

218

Views

Find $\theta \neq \sqrt{3}+\sqrt{5}$ such that $\mathbb{Q}(\theta) = \mathbb{Q}(\sqrt{3},\sqrt{5})$. Need a hint to get started.

Published on 16 Apr 2026 - 8:40

score 208 · Answer 11 · 2026-04-15 06:04:16

208

Views

Show that the extension $F \vert \mathbb{Q}$ is Galois

Published on 15 Apr 2026 - 6:04

score 5.4k · Answer 12 · 2026-04-15 06:47:06

Let $K$ be a field and $f(x)\in K[X]$ be a polynomial of degree $n$. And let $F$ be its splitting field. Show that $[F:K]$ divides $n!$.

score 52 · Answer 13 · 2026-04-14 22:28:16

52

Views

If $E$ is finite, show that $E=F(u)$ for some $u\in E$.

Published on 14 Apr 2026 - 22:28

score 58 · Answer 14 · 2026-04-14 18:09:58

58

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Prove that $e^{2\pi i/5}$ is not in the $7$-th cyclotomic field.

Published on 14 Apr 2026 - 18:09

score 437 · Answer 15 · 2026-04-15 00:22:31

437

Views

All possible degree 3 field extensions

Published on 15 Apr 2026 - 0:22

List Question

Extensions of the quadratic closure of $\mathbb{Q}$

Degree of field extension over a fixed field

$K/\mathbb{Q}$ contains a complex number - is complex conjugation in $\text{Aut}(K/\mathbb{Q})$?

Example of a field extension $K/F$ such that $K$ is the splitting field of some (non-separable) polynomial in $F$, but not Galois over $F$?

On the degree of cyclotomic fields extension.

For $a_{i}:=\sqrt{2+a_{i-1}}$, how to prove that $\mathbb{Q}\subset\mathbb{Q}(a_i)$ is cyclic of degree $2^i$?

Galois theory for beginners, mistake?

Show that a polynomial is still irreducible in a extension field

Degree of the extension field

Find $\theta \neq \sqrt{3}+\sqrt{5}$ such that $\mathbb{Q}(\theta) = \mathbb{Q}(\sqrt{3},\sqrt{5})$. Need a hint to get started.

Show that the extension $F \vert \mathbb{Q}$ is Galois

Let $K$ be a field and $f(x)\in K[X]$ be a polynomial of degree $n$. And let $F$ be its splitting field. Show that $[F:K]$ divides $n!$.

If $E$ is finite, show that $E=F(u)$ for some $u\in E$.

Prove that $e^{2\pi i/5}$ is not in the $7$-th cyclotomic field.

All possible degree 3 field extensions

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