Question

prove that $2 < [\bar{F} : F] < \infty$ is impossible

score 40 · Answer 1 · 2026-04-15 11:35:45

40

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prove that $2 < [\bar{F} : F] < \infty$ is impossible

Published on 15 Apr 2026 - 11:35

score 763 · Answer 2 · 2026-04-17 04:15:20

763

Views

Let $F$ be a field of characteristic $p > 0$, and let $K$ be a purely inseparable extension of F with $[K : F] = p^n$. Prove that $a^{p^n}\in F$

Published on 17 Apr 2026 - 4:15

score 1.9k · Answer 3 · 2026-04-16 06:30:09

1.9k

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Let $K$ and $L$ be extensions of $F$. Show that $KL$ is Galois over $F$ if both $K$ and $L$ are Galois over $F$. Is the converse true?

Published on 16 Apr 2026 - 6:30

score 905 · Answer 4 · 2026-04-13 16:38:38

905

Views

A counterexample to [LM:K] = [L:K] [M:K]

Published on 13 Apr 2026 - 16:38

score 989 · Answer 5 · 2026-04-11 15:00:45

989

Views

Let $k^p = k$, $K = k(x)$ and $F = k(u)$ for some $u \in K$. Show that $K/F$ is separable $\iff u \not\in K^p$.

Published on 11 Apr 2026 - 15:00

score 1k · Answer 6 · 2026-03-26 16:05:50

1k

Views

Classifying prime ideals of $\mathbb{Z}[i]$

Published on 26 Mar 2026 - 16:05

score 3.4k · Answer 7 · 2026-04-12 01:21:05

3.4k

Views

Subfields of a cyclotomic field

Published on 12 Apr 2026 - 1:21

score 68 · Answer 8 · 2026-04-16 14:32:37

68

Views

To calculate $[\mathbb Q(\sum_{j=1}^ka_j^{1/n_j}) : \mathbb Q]$ , when $a_i , n_i$ are positive integers with $a_1^{1/n_1}$ not an integer

Published on 16 Apr 2026 - 14:32

score 220 · Answer 9 · 2026-04-11 06:28:52

220

Views

Field extension and homomorphism between rings

Published on 11 Apr 2026 - 6:28

score 249 · Answer 10 · 2026-04-16 19:15:41

249

Views

Galois theory proving $\mathrm{Gal}(E/K_1 \cap K_2)= \left<\mathrm{Gal}(E/K_1),\mathrm{Gal}(E/K_2)\right>$

Published on 16 Apr 2026 - 19:15

score 78 · Answer 11 · 2026-04-14 03:16:58

78

Views

Why are the rationals fixed by this monomorphism?

Published on 14 Apr 2026 - 3:16

score 107 · Answer 12 · 2026-04-15 09:31:20

107

Views

Galois group of $\mathbb{C}(x(1-y),y(1-x)) \subset \mathbb{C}(x,y)$

Published on 15 Apr 2026 - 9:31

score 49 · Answer 13 · 2026-04-14 03:47:24

49

Views

Extension field adjointing a set

Published on 14 Apr 2026 - 3:47

score 603 · Answer 14 · 2026-03-25 22:03:28

603

Views

How to reduce a polynomial over an extension field (with or without MAGMA)

Published on 25 Mar 2026 - 22:03

score 498 · Answer 15 · 2026-04-13 03:58:39

498

Views

Ordered Field Extension

Published on 13 Apr 2026 - 3:58

List Question

prove that $2 < [\bar{F} : F] < \infty$ is impossible

Let $F$ be a field of characteristic $p > 0$, and let $K$ be a purely inseparable extension of F with $[K : F] = p^n$. Prove that $a^{p^n}\in F$

Let $K$ and $L$ be extensions of $F$. Show that $KL$ is Galois over $F$ if both $K$ and $L$ are Galois over $F$. Is the converse true?

A counterexample to [LM:K] = [L:K] [M:K]

Let $k^p = k$, $K = k(x)$ and $F = k(u)$ for some $u \in K$. Show that $K/F$ is separable $\iff u \not\in K^p$.

Classifying prime ideals of $\mathbb{Z}[i]$

Subfields of a cyclotomic field

To calculate $[\mathbb Q(\sum_{j=1}^ka_j^{1/n_j}) : \mathbb Q]$ , when $a_i , n_i$ are positive integers with $a_1^{1/n_1}$ not an integer

Field extension and homomorphism between rings

Galois theory proving $\mathrm{Gal}(E/K_1 \cap K_2)= \left<\mathrm{Gal}(E/K_1),\mathrm{Gal}(E/K_2)\right>$

Why are the rationals fixed by this monomorphism?

Galois group of $\mathbb{C}(x(1-y),y(1-x)) \subset \mathbb{C}(x,y)$

Extension field adjointing a set

How to reduce a polynomial over an extension field (with or without MAGMA)

Ordered Field Extension

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