List questions from extension-field

Published on 10 Apr 2026 - 19:05

935

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The minimal polynomial can't have multiple roots (in my proof)

Published on 12 Apr 2026 - 15:40

218

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Degree of $\mathbb{Q}(\sqrt{1 + \sqrt{2}})$ over $\mathbb{Q}$

Published on 16 Apr 2026 - 5:59

265

Views

Question about finite field extensions

Published on 13 Apr 2026 - 21:43

512

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Finite Galois extensions of the form $\frac{\mathbb Z_p[x]}{\langle p(x)\rangle}:\mathbb Z_p$

Published on 13 Apr 2026 - 12:16

68

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Symmetric elements in $k(x_1,y_1,\ldots,x_n,y_n)$ w.r.t. $\alpha_n: x_i \leftrightarrow y_i$

Published on 16 Apr 2026 - 18:10

8.4k

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$K$ is a splitting field $\iff$ any irreducible polynomial with a root in $K$ splits completely over $K$.

Published on 15 Apr 2026 - 17:23

170

Views

Prove the extension $\frac{\mathbb Z_7[x]}{\langle 1+x+x^3\rangle}:\mathbb Z_7$ is normal

Published on 14 Apr 2026 - 22:59

49

Views

A question on integral and algebraic elements

Published on 09 Apr 2026 - 16:21

813

Views

$E/K$ is field extension. $L_1,L_2$ are intermediate fields. $[L_1L_2:K]=[L_1:K][L_2:K]$ implies $K=L_1\cap L_2$.

Published on 16 Apr 2026 - 23:20

351

Views

$K/k$ finite extension, then $K(X)/k(X)$ is finite

Published on 16 Apr 2026 - 7:49

95

Views

Why is there a "maximal" intermediate field?

Published on 13 Apr 2026 - 11:14

228

Views

Simple proof of existence of number field extension with the same roots of unity?

Published on 14 Apr 2026 - 17:11

284

Views

If $\mathbb{F}(\alpha)$ and $\mathbb{F}(\beta)$ are isomorphic, then $\alpha\mapsto\beta$ is necessarily an isomorphism?

Published on 17 Apr 2026 - 11:30

92

Views

Two questions on irreducibilities of polynomials over an integral domain.

Published on 10 Apr 2026 - 2:18

List Question

About splitting fields

The minimal polynomial can't have multiple roots (in my proof)

Degree of $\mathbb{Q}(\sqrt{1 + \sqrt{2}})$ over $\mathbb{Q}$

Question about finite field extensions

Finite Galois extensions of the form $\frac{\mathbb Z_p[x]}{\langle p(x)\rangle}:\mathbb Z_p$

Symmetric elements in $k(x_1,y_1,\ldots,x_n,y_n)$ w.r.t. $\alpha_n: x_i \leftrightarrow y_i$

$K$ is a splitting field $\iff$ any irreducible polynomial with a root in $K$ splits completely over $K$.

Prove the extension $\frac{\mathbb Z_7[x]}{\langle 1+x+x^3\rangle}:\mathbb Z_7$ is normal

A question on integral and algebraic elements

$E/K$ is field extension. $L_1,L_2$ are intermediate fields. $[L_1L_2:K]=[L_1:K][L_2:K]$ implies $K=L_1\cap L_2$.

$K/k$ finite extension, then $K(X)/k(X)$ is finite

Why is there a "maximal" intermediate field?

Simple proof of existence of number field extension with the same roots of unity?

If $\mathbb{F}(\alpha)$ and $\mathbb{F}(\beta)$ are isomorphic, then $\alpha\mapsto\beta$ is necessarily an isomorphism?

Two questions on irreducibilities of polynomials over an integral domain.

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