Question

Let $K/F$ be a field extension. If $\alpha \in F(\alpha^m)$, $m > 1$, then $\alpha$ is algebraic in $F$.

score 85 · Answer 1 · 2019-08-17 04:14:47

85

Views

Let $K/F$ be a field extension. If $\alpha \in F(\alpha^m)$, $m > 1$, then $\alpha$ is algebraic in $F$.

Published on 17 Aug 2019 - 4:14

score 108 · Answer 2 · 2019-08-17 14:39:09

108

Views

Prove that $[\mathbb{Q}(\sqrt{3},i):\mathbb{Q}=4$. Find a number $a$, such that $\mathbb{Q}(a)=\mathbb{Q}(\sqrt{3},i)$

Published on 17 Aug 2019 - 14:39

score 221 · Answer 3 · 2019-08-19 08:12:48

221

Views

Is there an algebraic closed field which contains the complex field $\mathbb{C}$ strictly?

Published on 19 Aug 2019 - 8:12

score 159 · Answer 4 · 2019-08-19 09:27:22

159

Views

Show that $\mathbb{Q}(\sin\theta)$ is a field

Published on 19 Aug 2019 - 9:27

score 36 · Answer 5 · 2026-03-26 01:26:59

36

Views

Question about finite extensions and splitting fields

Published on 26 Mar 2026 - 1:26

score 87 · Answer 6 · 2019-08-19 23:47:39

87

Views

Chain of intermediate fields $Q ⊂ Q(α_ 8 ) ⊂ Q(α_ 4 ) ⊂ Q(α_ 2 ) ⊂ Q(α)$

Published on 19 Aug 2019 - 23:47

score 849 · Answer 7 · 2019-08-22 11:48:50

849

Views

Rational functions over $\mathbb{C}$ is algebraically closed

Published on 22 Aug 2019 - 11:48

score 334 · Answer 8 · 2026-03-26 01:29:17

334

Views

Does an algebraic closure of $F_p$ contain an element of infinite (multiplicative) order?

Published on 26 Mar 2026 - 1:29

score 37 · Answer 9 · 2026-03-29 20:34:00

37

Views

Show that $\operatorname{spec}(K[x_1,\dots,x_n])=\cup_{L/K}\operatorname{Im}(\psi_L)$ where $\psi_F(a)=\ker(f\mapsto f(a))$

Published on 29 Mar 2026 - 20:34

score 213 · Answer 10 · 2026-03-25 16:46:25

213

Views

Why is $[\mathbb{Q}(\zeta):\mathbb{Q}] = 8$ and not $14$? (Where $\zeta$ is a primitive $15^{th}$ root of unity)

Published on 25 Mar 2026 - 16:46

score 245 · Answer 11 · 2019-08-26 16:06:03

245

Views

If $[F(\alpha):F]=p$ and $[F(\beta):F]=q$, $p$ and $q$ distinct primes, then $[F(\alpha, \beta):F]=pq$

Published on 26 Aug 2019 - 16:06

score 95 · Answer 12 · 2026-03-26 01:29:11

95

Views

Why is the $\mathbb{Q}(i, \sqrt{3}, \sqrt[3]{5}) $splitting of the polynomial $(x^3 - 5)(x^2 + 1)$ and not just $(x^3 - 5)$?

Published on 26 Mar 2026 - 1:29

score 135 · Answer 13 · 2019-08-28 18:19:53

135

Views

Question about functionally independence

Published on 28 Aug 2019 - 18:19

score 39 · Answer 14 · 2019-08-28 19:18:55

39

Views

Natural extension of basis for a field extension is a basis for tensor product

Published on 28 Aug 2019 - 19:18

score 69 · Answer 15 · 2019-08-30 03:49:20

69

Views

Simple question for the Galois extension over $\mathbb{Q}$

Published on 30 Aug 2019 - 3:49

List Question

Let $K/F$ be a field extension. If $\alpha \in F(\alpha^m)$, $m > 1$, then $\alpha$ is algebraic in $F$.

Prove that $[\mathbb{Q}(\sqrt{3},i):\mathbb{Q}=4$. Find a number $a$, such that $\mathbb{Q}(a)=\mathbb{Q}(\sqrt{3},i)$

Is there an algebraic closed field which contains the complex field $\mathbb{C}$ strictly?

Show that $\mathbb{Q}(\sin\theta)$ is a field

Question about finite extensions and splitting fields

Chain of intermediate fields $Q ⊂ Q(α_ 8 ) ⊂ Q(α_ 4 ) ⊂ Q(α_ 2 ) ⊂ Q(α)$

Rational functions over $\mathbb{C}$ is algebraically closed

Does an algebraic closure of $F_p$ contain an element of infinite (multiplicative) order?

Show that $\operatorname{spec}(K[x_1,\dots,x_n])=\cup_{L/K}\operatorname{Im}(\psi_L)$ where $\psi_F(a)=\ker(f\mapsto f(a))$

Why is $[\mathbb{Q}(\zeta):\mathbb{Q}] = 8$ and not $14$? (Where $\zeta$ is a primitive $15^{th}$ root of unity)

If $[F(\alpha):F]=p$ and $[F(\beta):F]=q$, $p$ and $q$ distinct primes, then $[F(\alpha, \beta):F]=pq$

Why is the $\mathbb{Q}(i, \sqrt{3}, \sqrt[3]{5}) $splitting of the polynomial $(x^3 - 5)(x^2 + 1)$ and not just $(x^3 - 5)$?

Question about functionally independence

Natural extension of basis for a field extension is a basis for tensor product

Simple question for the Galois extension over $\mathbb{Q}$

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