Question

Are the $\{\alpha \in \mathbb{Q}(\theta) | Tr(\alpha\cdot\mathbb{Z}[\theta]) \subset \mathbb{Z}\}$ the algebraic integers of $\mathbb{Q}(\theta)$?

score 111 · Answer 1 · 2026-04-13 22:15:16

111

Views

Are the $\{\alpha \in \mathbb{Q}(\theta) | Tr(\alpha\cdot\mathbb{Z}[\theta]) \subset \mathbb{Z}\}$ the algebraic integers of $\mathbb{Q}(\theta)$?

Published on 13 Apr 2026 - 22:15

score 44 · Answer 2 · 2026-04-12 10:41:55

44

Views

Find all $\mathfrak{p}\in \operatorname{Spec}(\mathbb{Z}[\zeta])$ such that $\mathfrak{p}\cap\mathbb{Z}=7\mathbb{Z}$ where $\zeta=e^{(i\pi/3)}$

Published on 12 Apr 2026 - 10:41

score 67 · Answer 3 · 2026-04-15 17:36:58

67

Views

Show that a Galois extension is always generated by a square in characteristic 0

Published on 15 Apr 2026 - 17:36

score 64 · Answer 4 · 2026-03-29 14:03:11

64

Views

Prove that $M=E(\sqrt{a-b\sqrt{c}})$ is the minimal Galois extension of $F$ .

Published on 29 Mar 2026 - 14:03

score 71 · Answer 5 · 2026-04-17 12:00:38

71

Views

Examples of over $\Bbb Q(i)$ such that the Galois group is (i)$\Bbb Z_2\times \Bbb Z_2$ (ii) $D_4$

Published on 17 Apr 2026 - 12:00

score 146 · Answer 6 · 2026-04-15 06:15:52

146

Views

If $K = \mathbb{Q}(\alpha)$, then must $\mathcal{O}_K$ contain $\alpha$, and conversely?

Published on 15 Apr 2026 - 6:15

score 59 · Answer 7 · 2026-04-23 06:07:42

59

Views

Understanding $V=K\otimes_{\mathbb{F}_p}\mathbb{F}_{p^n}$ where $K$ is the algebraic closure of $\mathbb{F}_p$

Published on 23 Apr 2026 - 6:07

score 335 · Answer 8 · 2026-04-14 17:23:13

335

Views

Is $f(x) = x^{10}-x^5+1$ solvable by radicals?

Published on 14 Apr 2026 - 17:23

score 232 · Answer 9 · 2026-03-26 18:49:54

232

Views

Injective K-Homomorphisms are K-Automorphisms

Published on 26 Mar 2026 - 18:49

score 2k · Answer 10 · 2026-04-11 01:23:47

2k

Views

Field Extension with Galois group $S_n$ is the splitting field of a polynomial of degree $n$

Published on 11 Apr 2026 - 1:23

score 308 · Answer 11 · 2026-04-09 08:24:25

308

Views

Questions regarding tower of normal/separable extensions

Published on 09 Apr 2026 - 8:24

score 148 · Answer 12 · 2026-04-23 06:10:07

148

Views

Proving $F(\alpha)/F$ is a Galois extension, where $\alpha \in E \supset F$ is a root of $x^p-x-a=p(x) \in F[X]$, irreducible in $F$

Published on 23 Apr 2026 - 6:10

score 53 · Answer 13 · 2026-04-15 00:04:13

53

Views

Show that if $[E:F]$ is finite then $[E(u):E]\leq [F(u):F]$ is a finite extension

Published on 15 Apr 2026 - 0:04

score 666 · Answer 14 · 2026-03-29 14:02:54

666

Views

Prove that $\mathbb{Q}\left(\sqrt{(2+\sqrt{2})(3+\sqrt{3})}\right)$ is Galois over $\mathbb{Q}$ and determine the Galois group

Published on 29 Mar 2026 - 14:02

score 72 · Answer 15 · 2026-04-16 08:11:51

72

Views

Show that $x^2+x+1$ is irreducible over $GF(2^n)$ for each odd $n$.

Published on 16 Apr 2026 - 8:11

List Question

Are the $\{\alpha \in \mathbb{Q}(\theta) | Tr(\alpha\cdot\mathbb{Z}[\theta]) \subset \mathbb{Z}\}$ the algebraic integers of $\mathbb{Q}(\theta)$?

Find all $\mathfrak{p}\in \operatorname{Spec}(\mathbb{Z}[\zeta])$ such that $\mathfrak{p}\cap\mathbb{Z}=7\mathbb{Z}$ where $\zeta=e^{(i\pi/3)}$

Show that a Galois extension is always generated by a square in characteristic 0

Prove that $M=E(\sqrt{a-b\sqrt{c}})$ is the minimal Galois extension of $F$ .

Examples of over $\Bbb Q(i)$ such that the Galois group is (i)$\Bbb Z_2\times \Bbb Z_2$ (ii) $D_4$

If $K = \mathbb{Q}(\alpha)$, then must $\mathcal{O}_K$ contain $\alpha$, and conversely?

Understanding $V=K\otimes_{\mathbb{F}_p}\mathbb{F}_{p^n}$ where $K$ is the algebraic closure of $\mathbb{F}_p$

Is $f(x) = x^{10}-x^5+1$ solvable by radicals?

Injective K-Homomorphisms are K-Automorphisms

Field Extension with Galois group $S_n$ is the splitting field of a polynomial of degree $n$

Questions regarding tower of normal/separable extensions

Proving $F(\alpha)/F$ is a Galois extension, where $\alpha \in E \supset F$ is a root of $x^p-x-a=p(x) \in F[X]$, irreducible in $F$

Show that if $[E:F]$ is finite then $[E(u):E]\leq [F(u):F]$ is a finite extension

Prove that $\mathbb{Q}\left(\sqrt{(2+\sqrt{2})(3+\sqrt{3})}\right)$ is Galois over $\mathbb{Q}$ and determine the Galois group

Show that $x^2+x+1$ is irreducible over $GF(2^n)$ for each odd $n$.

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