Question

If $K(\alpha)/K$ is Galois and $\exists \sigma\in G$ such that $\sigma \alpha=\alpha^{-1}$, show $[K(\alpha+\alpha^{-1}):K]=\frac{1}{2}[K(\alpha):K]$.

score 110 · Answer 1 · 2018-08-10 20:06:15

110

Views

If $K(\alpha)/K$ is Galois and $\exists \sigma\in G$ such that $\sigma \alpha=\alpha^{-1}$, show $[K(\alpha+\alpha^{-1}):K]=\frac{1}{2}[K(\alpha):K]$.

Published on 10 Aug 2018 - 20:06

score 397 · Answer 2 · 2018-08-11 02:54:52

397

Views

Given a field extension with prime degree, if $\operatorname{Aut}(K/F) > 1$, then this extension is Galois?

Published on 11 Aug 2018 - 2:54

score 354 · Answer 3 · 2018-08-12 05:08:26

354

Views

If $F\subset D \subset E$ then $D$ must be a field

Published on 12 Aug 2018 - 5:08

score 704 · Answer 4 · 2018-08-16 00:48:29

704

Views

Is $F_p (X,Y)$ a simple extension over $F_p (X^p,Y^p)$?

Published on 16 Aug 2018 - 0:48

score 357 · Answer 5 · 2026-03-25 14:47:01

357

Views

Showing that the Galois group has an element of order 8.

Published on 25 Mar 2026 - 14:47

score 84 · Answer 6 · 2026-03-25 17:22:56

84

Views

Are finite extensions of global function fields always simple?

Published on 25 Mar 2026 - 17:22

score 34 · Answer 7 · 2018-08-18 06:54:14

34

Views

Given a field extension E/F, $f(x)\in E[x]$, and $f(\alpha )\in F, \forall \alpha \in F$. Does $f(x)\in F[x]$?

Published on 18 Aug 2018 - 6:54

score 41 · Answer 8 · 2026-03-25 11:04:46

41

Views

How to obtain the degree of this field extension?

Published on 25 Mar 2026 - 11:04

score 222 · Answer 9 · 2026-03-27 08:39:00

222

Views

Purely Inseparable Field Extension Differentials

Published on 27 Mar 2026 - 8:39

score 67 · Answer 10 · 2018-08-19 23:22:45

67

Views

Is $\mathbb{Z}_p(y)$ a field, where $y$ is an indeterminate/placeholder?

Published on 19 Aug 2018 - 23:22

score 991 · Answer 11 · 2026-03-27 13:24:24

991

Views

Galois group of $x^3-3x-1$ over $\mathbb{Q}$

Published on 27 Mar 2026 - 13:24

score 90 · Answer 12 · 2026-03-25 12:46:16

90

Views

Squaring the circle, compass-and-straightedge construction

Published on 25 Mar 2026 - 12:46

score 123 · Answer 13 · 2026-03-26 07:59:19

123

Views

Sums of powers of algebraic numbers that are rational

Published on 26 Mar 2026 - 7:59

score 629 · Answer 14 · 2018-08-24 10:10:08

629

Views

Prove or disprove that $[L:K] = \deg(p)$

Published on 24 Aug 2018 - 10:10

score 787 · Answer 15 · 2026-03-25 16:13:19

787

Views

Subfields of the $p$-adic numbers

Published on 25 Mar 2026 - 16:13

List Question

If $K(\alpha)/K$ is Galois and $\exists \sigma\in G$ such that $\sigma \alpha=\alpha^{-1}$, show $[K(\alpha+\alpha^{-1}):K]=\frac{1}{2}[K(\alpha):K]$.

Given a field extension with prime degree, if $\operatorname{Aut}(K/F) > 1$, then this extension is Galois?

If $F\subset D \subset E$ then $D$ must be a field

Is $F_p (X,Y)$ a simple extension over $F_p (X^p,Y^p)$?

Showing that the Galois group has an element of order 8.

Are finite extensions of global function fields always simple?

Given a field extension E/F, $f(x)\in E[x]$, and $f(\alpha )\in F, \forall \alpha \in F$. Does $f(x)\in F[x]$?

How to obtain the degree of this field extension?

Purely Inseparable Field Extension Differentials

Is $\mathbb{Z}_p(y)$ a field, where $y$ is an indeterminate/placeholder?

Galois group of $x^3-3x-1$ over $\mathbb{Q}$

Squaring the circle, compass-and-straightedge construction

Sums of powers of algebraic numbers that are rational

Prove or disprove that $[L:K] = \deg(p)$

Subfields of the $p$-adic numbers

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