Question

How to formally conclude that $ x^2-(\zeta_9+\zeta^{-1}_9)x+1 $ is irreducible over $\Bbb Q(\zeta_9+\zeta^{-1}_9)$

score 97 · Answer 1 · 2026-04-14 21:15:54

97

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How to formally conclude that $ x^2-(\zeta_9+\zeta^{-1}_9)x+1 $ is irreducible over $\Bbb Q(\zeta_9+\zeta^{-1}_9)$

Published on 14 Apr 2026 - 21:15

score 173 · Answer 2 · 2026-04-16 05:37:48

173

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Find the monic polynomial of $\sqrt[8]{7}$ over each of the following fields

Published on 16 Apr 2026 - 5:37

score 39 · Answer 3 · 2026-04-10 17:45:45

39

Views

Question about transcendental extension

Published on 10 Apr 2026 - 17:45

score 1.2k · Answer 4 · 2026-04-16 07:41:14

1.2k

Views

Prove that the splitting field of $X^n - 1$ has degree $\phi(n)$ over $\mathbb{Q}$

Published on 16 Apr 2026 - 7:41

score 1.5k · Answer 5 · 2026-04-17 12:12:21

1.5k

Views

Describe the elements of the extension $\mathbb{Q}(2^{1/4})$ over the field $\mathbb{Q}(\sqrt{2})$

Published on 17 Apr 2026 - 12:12

score 1.1k · Answer 6 · 2026-04-11 12:13:41

1.1k

Views

Composite of two Galois extensions

Published on 11 Apr 2026 - 12:13

score 155 · Answer 7 · 2026-04-16 12:50:34

155

Views

How to formally prove that the degree of $\Bbb Q（\sqrt[5]{6},\sqrt[3]{7}）$ is $15$？

Published on 16 Apr 2026 - 12:50

score 242 · Answer 8 · 2026-04-15 23:51:58

242

Views

Degree of extension $\Bbb Q(\sqrt[4]{5},\sqrt[6]{7})$.

Published on 15 Apr 2026 - 23:51

score 595 · Answer 9 · 2026-04-15 20:21:58

595

Views

Find the minimal polynomial of $\zeta_9+\zeta^{-1}_9$ over $\Bbb Q$

Published on 15 Apr 2026 - 20:21

score 82 · Answer 10 · 2026-04-09 21:00:33

82

Views

How do I approach this? (Galois Theory)

Published on 09 Apr 2026 - 21:00

score 41 · Answer 11 · 2026-04-13 23:31:46

41

Views

Question regarding field extension and Galois Group.

Published on 13 Apr 2026 - 23:31

score 658 · Answer 12 · 2026-04-14 05:57:31

658

Views

Difference between "algebraically closed field" and the "relatively algebraically closed field"?

Published on 14 Apr 2026 - 5:57

score 112 · Answer 13 · 2026-04-16 01:09:44

112

Views

(Proof Explanation) $\Bbb Q(\sqrt d)$ is never isomorphic to $\Bbb Q(\sqrt {d'})$ as fields

Published on 16 Apr 2026 - 1:09

score 5k · Answer 14 · 2026-04-13 18:32:19

5k

Views

Galois extension of Q(roots of unity) over Q

Published on 13 Apr 2026 - 18:32

score 61 · Answer 15 · 2026-04-15 13:38:16

61

Views

(Proof Verification) Prove that $f(x)$ is irreducible over $K$ implies it is irreducible over $F$.

Published on 15 Apr 2026 - 13:38

List Question

How to formally conclude that $ x^2-(\zeta_9+\zeta^{-1}_9)x+1 $ is irreducible over $\Bbb Q(\zeta_9+\zeta^{-1}_9)$

Find the monic polynomial of $\sqrt[8]{7}$ over each of the following fields

Question about transcendental extension

Prove that the splitting field of $X^n - 1$ has degree $\phi(n)$ over $\mathbb{Q}$

Describe the elements of the extension $\mathbb{Q}(2^{1/4})$ over the field $\mathbb{Q}(\sqrt{2})$

Composite of two Galois extensions

How to formally prove that the degree of $\Bbb Q（\sqrt[5]{6},\sqrt[3]{7}）$ is $15$？

Degree of extension $\Bbb Q(\sqrt[4]{5},\sqrt[6]{7})$.

Find the minimal polynomial of $\zeta_9+\zeta^{-1}_9$ over $\Bbb Q$

How do I approach this? (Galois Theory)

Question regarding field extension and Galois Group.

Difference between "algebraically closed field" and the "relatively algebraically closed field"?

(Proof Explanation) $\Bbb Q(\sqrt d)$ is never isomorphic to $\Bbb Q(\sqrt {d'})$ as fields

Galois extension of Q(roots of unity) over Q

(Proof Verification) Prove that $f(x)$ is irreducible over $K$ implies it is irreducible over $F$.

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