Question

Degree of $\mathbb{Q}(\sqrt[3]{2},\sqrt{2})$ over $\mathbb{Q}$

score 108 · Answer 1 · 2026-04-16 03:01:09

108

Views

Degree of $\mathbb{Q}(\sqrt[3]{2},\sqrt{2})$ over $\mathbb{Q}$

Published on 16 Apr 2026 - 3:01

score 26 · Answer 2 · 2026-04-14 17:39:21

26

Views

Let $L|K$ be a field extension and $\alpha \in L$ separable. Why does $\sigma(\mu_{\alpha})$ factor into pairwise different factors in $\bar{L}?$

Published on 14 Apr 2026 - 17:39

score 334 · Answer 3 · 2026-04-14 14:37:32

334

Views

Inverse of an algebraic element over finite extension

Published on 14 Apr 2026 - 14:37

score 307 · Answer 4 · 2026-04-16 16:20:42

307

Views

$|F|=q$ (power of a prime) , $|F(u)|=q^a , |F(v)|=q^b$ , and $F=F(u) \cap F(v)$ , then is it true that $|F(u) \cap F(v)|=q^{gcd(a,b)}$ ?

Published on 16 Apr 2026 - 16:20

score 52 · Answer 5 · 2026-04-19 16:40:12

52

Views

Roots of an equation with coefficients in $\mathbb{Z}_{19}$

Published on 19 Apr 2026 - 16:40

score 289 · Answer 6 · 2026-04-15 10:41:42

289

Views

$\mathbb{Z}[\alpha]$ is f.g. if $\alpha$ is an algebraic integer.

Published on 15 Apr 2026 - 10:41

score 80 · Answer 7 · 2026-04-16 20:00:16

80

Views

Prove that $\mathbb{Q}[\sqrt{2}] \subsetneqq\mathbb{Q}[\sqrt{2}+\sqrt{3}] $

Published on 16 Apr 2026 - 20:00

score 163 · Answer 8 · 2026-04-15 02:25:50

163

Views

Smallest normal extension of F containing E is composite of E and all its conjugates over F. Proof

Published on 15 Apr 2026 - 2:25

score 198 · Answer 9 · 2026-04-15 17:07:17

198

Views

Basis for $\mathbb{Q}(\alpha, \beta)$ over $\mathbb{Q}$

Published on 15 Apr 2026 - 17:07

score 462 · Answer 10 · 2026-04-13 15:00:46

462

Views

Generators of $\mathbb{Z}[\alpha.\beta]$ and $\mathbb{Z}[\alpha\pm\beta]$ when $\mathbb{Z}[\alpha]$, $\mathbb{Z}[\beta]$ are f.g.

Published on 13 Apr 2026 - 15:00

score 179 · Answer 11 · 2026-04-14 18:17:56

179

Views

Given a number field $K$, showing that $\mathbb{Q}O_K=K$

Published on 14 Apr 2026 - 18:17

score 443 · Answer 12 · 2026-04-16 08:10:36

443

Views

Nonstandard ordering on $\mathbb Q(\sqrt 2)$

Published on 16 Apr 2026 - 8:10

score 228 · Answer 13 · 2026-04-16 08:04:16

228

Views

Show that $ F_{7}[i]=\{a+bi : a,b \in F_{7} \}$ is a field but that $F_5[i]$ is not

Published on 16 Apr 2026 - 8:04

score 49 · Answer 14 · 2026-04-13 02:54:34

49

Views

Multiplying a field by a constant

Published on 13 Apr 2026 - 2:54

score 206 · Answer 15 · 2026-04-16 10:45:46

206

Views

If we take $R$ to be the field of real numbers. Show that $R[x]/\langle x^2 + 1\rangle$ is a field.

Published on 16 Apr 2026 - 10:45

List Question

Degree of $\mathbb{Q}(\sqrt[3]{2},\sqrt{2})$ over $\mathbb{Q}$

Let $L|K$ be a field extension and $\alpha \in L$ separable. Why does $\sigma(\mu_{\alpha})$ factor into pairwise different factors in $\bar{L}?$

Inverse of an algebraic element over finite extension

$|F|=q$ (power of a prime) , $|F(u)|=q^a , |F(v)|=q^b$ , and $F=F(u) \cap F(v)$ , then is it true that $|F(u) \cap F(v)|=q^{gcd(a,b)}$ ?

Roots of an equation with coefficients in $\mathbb{Z}_{19}$

$\mathbb{Z}[\alpha]$ is f.g. if $\alpha$ is an algebraic integer.

Prove that $\mathbb{Q}[\sqrt{2}] \subsetneqq\mathbb{Q}[\sqrt{2}+\sqrt{3}] $

Smallest normal extension of F containing E is composite of E and all its conjugates over F. Proof

Basis for $\mathbb{Q}(\alpha, \beta)$ over $\mathbb{Q}$

Generators of $\mathbb{Z}[\alpha.\beta]$ and $\mathbb{Z}[\alpha\pm\beta]$ when $\mathbb{Z}[\alpha]$, $\mathbb{Z}[\beta]$ are f.g.

Given a number field $K$, showing that $\mathbb{Q}O_K=K$

Nonstandard ordering on $\mathbb Q(\sqrt 2)$

Show that $ F_{7}[i]=\{a+bi : a,b \in F_{7} \}$ is a field but that $F_5[i]$ is not

Multiplying a field by a constant

If we take $R$ to be the field of real numbers. Show that $R[x]/\langle x^2 + 1\rangle$ is a field.

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