Question

Show that $\mathbb{Q}(\sqrt{5}+\sqrt[3]{2})=\mathbb{Q}(\sqrt{5},\sqrt[3]{2})$.

score 2.5k · Answer 1 · 2026-04-05 15:15:14

2.5k

Views

Show that $\mathbb{Q}(\sqrt{5}+\sqrt[3]{2})=\mathbb{Q}(\sqrt{5},\sqrt[3]{2})$.

Published on 05 Apr 2026 - 15:15

score 461 · Answer 2 · 2015-06-25 14:44:43

461

Views

Degree of minimal polynomial of the sum of two algebraic elements over $\mathbb Q$

Published on 25 Jun 2015 - 14:44

score 1.3k · Answer 3 · 2015-06-25 23:23:38

1.3k

Views

Integral basis of $\mathbb{Q}(\theta)$, where $\theta^3-\theta-4=0$

Published on 25 Jun 2015 - 23:23

score 891 · Answer 4 · 2015-06-26 09:16:31

891

Views

Isomorphism of quadratic extensions (of a number field)

Published on 26 Jun 2015 - 9:16

score 57 · Answer 5 · 2015-06-28 03:47:43

57

Views

Intermediate "prime" extensions [Rotman]

Published on 28 Jun 2015 - 3:47

score 301 · Answer 6 · 2026-03-27 22:11:56

301

Views

Irreducible polynomial over $\mathbb{Q}(\zeta)$

Published on 27 Mar 2026 - 22:11

score 1.4k · Answer 7 · 2015-07-05 09:33:28

1.4k

Views

If $F(\alpha)=F(\beta)$, must $\alpha$ and $\beta$ have the same minimal polynomial?

Published on 05 Jul 2015 - 9:33

score 725 · Answer 8 · 2026-04-11 16:14:59

725

Views

Radical extension and algebraic solution of an irreducible polynomial

Published on 11 Apr 2026 - 16:14

score 72 · Answer 9 · 2015-07-10 05:01:54

72

Views

dim. of $\left[\mathbb{Q}\left(\sqrt[n]{a},\zeta_{n}\right):\mathbb{Q}\right]$

Published on 10 Jul 2015 - 5:01

score 55 · Answer 10 · 2015-07-10 05:55:16

"If $\text{Gal}\left(K/F\right)=\left<\sigma\right>$, $N_{K/F}\left(\alpha\right)=1$ , then $\alpha=\frac{\beta}{\sigma\left(\beta\right)}$."

score 177 · Answer 11 · 2015-07-13 16:05:31

177

Views

Given $K(\alpha)/K$ and $K(\beta)/K$ abelian extensions, prove that $K(\alpha + \beta)/K$ is an abelian extension.

Published on 13 Jul 2015 - 16:05

score 54 · Answer 12 · 2026-04-11 22:14:50

54

Views

Uses for coefficients other than the trace and norm for an element belonging to a field.

Published on 11 Apr 2026 - 22:14

score 416 · Answer 13 · 2015-07-15 01:51:16

416

Views

Example check: two algebraically closed fields with one a subset of the other

Published on 15 Jul 2015 - 1:51

score 430 · Answer 14 · 2026-04-11 14:58:30

430

Views

Galois group and degree of splitting field over complex rational functions.

Published on 11 Apr 2026 - 14:58

score 1.1k · Answer 15 · 2026-04-11 20:32:50

1.1k

Views

Field extension generated by $\alpha$ and separability

Published on 11 Apr 2026 - 20:32

List Question

Show that $\mathbb{Q}(\sqrt{5}+\sqrt[3]{2})=\mathbb{Q}(\sqrt{5},\sqrt[3]{2})$.

Degree of minimal polynomial of the sum of two algebraic elements over $\mathbb Q$

Integral basis of $\mathbb{Q}(\theta)$, where $\theta^3-\theta-4=0$

Isomorphism of quadratic extensions (of a number field)

Intermediate "prime" extensions [Rotman]

Irreducible polynomial over $\mathbb{Q}(\zeta)$

If $F(\alpha)=F(\beta)$, must $\alpha$ and $\beta$ have the same minimal polynomial?

Radical extension and algebraic solution of an irreducible polynomial

dim. of $\left[\mathbb{Q}\left(\sqrt[n]{a},\zeta_{n}\right):\mathbb{Q}\right]$

"If $\text{Gal}\left(K/F\right)=\left<\sigma\right>$, $N_{K/F}\left(\alpha\right)=1$ , then $\alpha=\frac{\beta}{\sigma\left(\beta\right)}$."

Given $K(\alpha)/K$ and $K(\beta)/K$ abelian extensions, prove that $K(\alpha + \beta)/K$ is an abelian extension.

Uses for coefficients other than the trace and norm for an element belonging to a field.

Example check: two algebraically closed fields with one a subset of the other

Galois group and degree of splitting field over complex rational functions.

Field extension generated by $\alpha$ and separability

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