Question

Let $\sigma \in\operatorname{Aut}(K)$ have infinite order and $F = \mathcal{F}(\sigma)$. Show that if $K/F$ is algebraic, then $K$ is normal over $F$.

score 529 · Answer 1 · 2026-03-25 18:57:50

529

Views

Let $\sigma \in\operatorname{Aut}(K)$ have infinite order and $F = \mathcal{F}(\sigma)$. Show that if $K/F$ is algebraic, then $K$ is normal over $F$.

Published on 25 Mar 2026 - 18:57

score 44 · Answer 2 · 2026-03-26 06:34:45

44

Views

Why do we have that $\Bbb F_{p^d}$ is a sub-field of $\Bbb F_{p^n}$ iff $d\mid n$?

Published on 26 Mar 2026 - 6:34

score 172 · Answer 3 · 2026-03-27 15:13:19

172

Views

Proving $\text{Gal}(K/F) \cong \text{Gal}(K/E_1) \times \text{Gal}(K/E_2)$

Published on 27 Mar 2026 - 15:13

score 143 · Answer 4 · 2026-03-25 06:22:13

143

Views

Showing that $\sqrt[3]{2}\notin\Bbb Q(\alpha_1,...,\alpha_k)$ where $\alpha_i^2\in\Bbb Q\ \forall i$

Published on 25 Mar 2026 - 6:22

score 530 · Answer 5 · 2026-03-17 08:02:21

530

Views

How does knowing that $\sqrt{7}\notin\mathbb{Q}(e^{2\pi i/7})$ help construct this character table?

Published on 17 Mar 2026 - 8:02

score 60 · Answer 6 · 2018-05-05 22:06:54

60

Views

If $x$ is an algebraical element over $F$, then $F(x) = F[x]$

Published on 05 May 2018 - 22:06

score 187 · Answer 7 · 2026-03-19 23:17:09

187

Views

Lüroth's theorem for transcendence degree two

Published on 19 Mar 2026 - 23:17

score 50 · Answer 8 · 2026-03-25 18:57:44

50

Views

degree/dimension of field extension if $K=\Bbb Q(\alpha)$ and $F=\Bbb Q$

Published on 25 Mar 2026 - 18:57

score 314 · Answer 9 · 2026-03-25 08:07:35

314

Views

Isomorphism between two extensions $\Bbb F_2(\alpha)$ and $\Bbb F_2(\beta)$

Published on 25 Mar 2026 - 8:07

score 305 · Answer 10 · 2026-03-25 18:59:09

305

Views

Galois extension over a field of characteristic $0$ of order $2^n$

Published on 25 Mar 2026 - 18:59

score 73 · Answer 11 · 2026-03-25 18:58:47

73

Views

Arguing that an extension is not normal

Published on 25 Mar 2026 - 18:58

score 76 · Answer 12 · 2018-05-07 02:49:38

76

Views

Consider $p(x)=x^4 -2x^2 -2 \in \mathbb{Q}[x]$.

Published on 07 May 2018 - 2:49

score 219 · Answer 13 · 2026-03-25 01:16:58

219

Views

Galois groups and groups of Dirichlet characters

Published on 25 Mar 2026 - 1:16

score 327 · Answer 14 · 2026-03-25 20:22:58

327

Views

Can the skew field of quaternions be seen as an extension over $\Bbb C$

Published on 25 Mar 2026 - 20:22

score 355 · Answer 15 · 2026-03-25 20:20:29

355

Views

Given a field, the field of algebraic elements over it in an algebraically closed field extension is again algebraically closed?

Published on 25 Mar 2026 - 20:20

List Question

Let $\sigma \in\operatorname{Aut}(K)$ have infinite order and $F = \mathcal{F}(\sigma)$. Show that if $K/F$ is algebraic, then $K$ is normal over $F$.

Why do we have that $\Bbb F_{p^d}$ is a sub-field of $\Bbb F_{p^n}$ iff $d\mid n$?

Proving $\text{Gal}(K/F) \cong \text{Gal}(K/E_1) \times \text{Gal}(K/E_2)$

Showing that $\sqrt[3]{2}\notin\Bbb Q(\alpha_1,...,\alpha_k)$ where $\alpha_i^2\in\Bbb Q\ \forall i$

How does knowing that $\sqrt{7}\notin\mathbb{Q}(e^{2\pi i/7})$ help construct this character table?

If $x$ is an algebraical element over $F$, then $F(x) = F[x]$

Lüroth's theorem for transcendence degree two

degree/dimension of field extension if $K=\Bbb Q(\alpha)$ and $F=\Bbb Q$

Isomorphism between two extensions $\Bbb F_2(\alpha)$ and $\Bbb F_2(\beta)$

Galois extension over a field of characteristic $0$ of order $2^n$

Arguing that an extension is not normal

Consider $p(x)=x^4 -2x^2 -2 \in \mathbb{Q}[x]$.

Galois groups and groups of Dirichlet characters

Can the skew field of quaternions be seen as an extension over $\Bbb C$

Given a field, the field of algebraic elements over it in an algebraically closed field extension is again algebraically closed?

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