Question

When is a field an algebraic extension of a fixed subfield?

score 226 · Answer 1 · 2026-04-12 00:50:41

226

Views

When is a field an algebraic extension of a fixed subfield?

Published on 12 Apr 2026 - 0:50

score 283 · Answer 2 · 2026-04-12 00:29:58

283

Views

Is an algebraic extension of a separably closed field separably closed?

Published on 12 Apr 2026 - 0:29

score 74 · Answer 3 · 2026-04-15 07:24:54

74

Views

If every ring $R$ such that $K \subset R \subset L$ is a field, then $L \supset K$ is an algebraic extension.

Published on 15 Apr 2026 - 7:24

score 264 · Answer 4 · 2026-04-17 07:59:12

264

Views

For $a,b \in \mathbb {F}_{{2^n}}$, $n$ is odd; $a^2+b^2+ab=0$ implies that $a=b=0$.

Published on 17 Apr 2026 - 7:59

score 107 · Answer 5 · 2026-04-13 22:09:02

107

Views

About algebraically independent sets

Published on 13 Apr 2026 - 22:09

score 254 · Answer 6 · 2026-04-11 05:22:04

254

Views

Algebraic closure of subfield is unique?

Published on 11 Apr 2026 - 5:22

score 194 · Answer 7 · 2026-04-15 21:19:29

194

Views

$K$ be a finite extension of $\mathbb Q$ then there exists a finite extension $E$ over $K$ such that $E/K$ is not normal.

Published on 15 Apr 2026 - 21:19

score 47 · Answer 8 · 2026-04-11 00:26:19

47

Views

Is the degree of splitting field remains same when we shift to an algebraic extension?

Published on 11 Apr 2026 - 0:26

score 64 · Answer 9 · 2026-04-15 23:17:24

64

Views

If $\alpha \in \mathbb Q[\zeta_n]$ satisfying $\alpha^m=2$ for some positive integer $m$, then $m=1$ or $2$.

Published on 15 Apr 2026 - 23:17

score 191 · Answer 10 · 2026-04-16 01:31:48

191

Views

For $n\in \mathbb N$, the splitting field of $x^n-2$ over $\mathbb Q$ has degree $n\cdot\phi(n)$.

Published on 16 Apr 2026 - 1:31

score 80 · Answer 11 · 2026-04-13 12:22:17

80

Views

degree of the extension of a function field

Published on 13 Apr 2026 - 12:22

score 125 · Answer 12 · 2026-04-16 04:48:04

125

Views

For $\omega=e^{2\pi i/3}$ and $\epsilon =e^{2\pi i/5}$, determine $|\mathbb Q[\omega +\epsilon]:\mathbb Q|$.

Published on 16 Apr 2026 - 4:48

score 47 · Answer 13 · 2026-04-17 14:35:09

47

Views

$|E:F|=1$ if $\operatorname{char}(k)=0$ and $p^m$ for some $m \in \mathbb N \cup \{0\}$ if $\operatorname{char}(k)=p$.

Published on 17 Apr 2026 - 14:35

score 246 · Answer 14 · 2026-04-16 07:24:29

246

Views

Field extension obtained by adjoining elements to base field

Published on 16 Apr 2026 - 7:24

score 75 · Answer 15 · 2026-04-17 04:47:20

75

Views

For $K=\mathbb Q(\alpha,\beta)$ with $\beta^2=\alpha^3$ if $\beta \in \mathbb Q(\alpha)$, then $|K:\mathbb Q|$ is finite.

Published on 17 Apr 2026 - 4:47

List Question

When is a field an algebraic extension of a fixed subfield?

Is an algebraic extension of a separably closed field separably closed?

If every ring $R$ such that $K \subset R \subset L$ is a field, then $L \supset K$ is an algebraic extension.

For $a,b \in \mathbb {F}_{{2^n}}$, $n$ is odd; $a^2+b^2+ab=0$ implies that $a=b=0$.

About algebraically independent sets

Algebraic closure of subfield is unique?

$K$ be a finite extension of $\mathbb Q$ then there exists a finite extension $E$ over $K$ such that $E/K$ is not normal.

Is the degree of splitting field remains same when we shift to an algebraic extension?

If $\alpha \in \mathbb Q[\zeta_n]$ satisfying $\alpha^m=2$ for some positive integer $m$, then $m=1$ or $2$.

For $n\in \mathbb N$, the splitting field of $x^n-2$ over $\mathbb Q$ has degree $n\cdot\phi(n)$.

degree of the extension of a function field

For $\omega=e^{2\pi i/3}$ and $\epsilon =e^{2\pi i/5}$, determine $|\mathbb Q[\omega +\epsilon]:\mathbb Q|$.

$|E:F|=1$ if $\operatorname{char}(k)=0$ and $p^m$ for some $m \in \mathbb N \cup \{0\}$ if $\operatorname{char}(k)=p$.

Field extension obtained by adjoining elements to base field

For $K=\mathbb Q(\alpha,\beta)$ with $\beta^2=\alpha^3$ if $\beta \in \mathbb Q(\alpha)$, then $|K:\mathbb Q|$ is finite.

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