List questions from field-theory

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If $a$ is algebraic over $F$ and $b$ is transcendental over $F$, then $a+b$ is transcendental over $F$.

Published on 26 Mar 2026 - 4:34

346

Views

Module and Noetherian/Artinian Rings

Published on 09 Apr 2026 - 16:12

361

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Criteria to check for subfield and subdomain

Published on 11 Apr 2026 - 10:21

249

Views

Can $\mathbb{Z}_{6}$ be a subring of some field?

Published on 10 Apr 2026 - 17:08

112

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Quadratic extension of quadratic extensions

Published on 14 Apr 2026 - 8:10

1.3k

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$K$ is an extension of $F$ and $a,b\in K-F$. If $ab$ is algebraic over $F$ then $b$ is algebraic over $F(a)$.

Published on 15 Apr 2026 - 11:43

30

Views

How to show $[F(a):F[a^4+a^2+1]$ is finite.

Published on 13 Apr 2026 - 7:51

259

Views

How to find field embeddings

Published on 09 Apr 2026 - 10:16

60

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$x^k = na$ has solution over the field $F, \text{char} F \neq 2$

Published on 14 Apr 2026 - 10:26

43

Views

Why the ideals here are in this form?

Published on 13 Apr 2026 - 9:11

251

Views

Good book for Local Fields/ Commutative algebra?

Published on 15 Apr 2026 - 3:30

692

Views

Why two extension fields are isomorphic as vector spaces but not fields?

Published on 16 Apr 2026 - 2:08

62

Views

Linear transformations of fields

Published on 13 Apr 2026 - 8:32

823

Views

Proof of an existence of an algebraic closure of a given field

Published on 11 Apr 2026 - 17:33

68

Views

Decompose $a = a_1\cdots a_k$ and $a_1 + \dots +a_k = 0$

Published on 14 Apr 2026 - 23:35

List Question

If $a$ is algebraic over $F$ and $b$ is transcendental over $F$, then $a+b$ is transcendental over $F$.

Module and Noetherian/Artinian Rings

Criteria to check for subfield and subdomain

Can $\mathbb{Z}_{6}$ be a subring of some field?

Quadratic extension of quadratic extensions

$K$ is an extension of $F$ and $a,b\in K-F$. If $ab$ is algebraic over $F$ then $b$ is algebraic over $F(a)$.

How to show $[F(a):F[a^4+a^2+1]$ is finite.

How to find field embeddings

$x^k = na$ has solution over the field $F, \text{char} F \neq 2$

Why the ideals here are in this form?

Good book for Local Fields/ Commutative algebra?

Why two extension fields are isomorphic as vector spaces but not fields?

Linear transformations of fields

Proof of an existence of an algebraic closure of a given field

Decompose $a = a_1\cdots a_k$ and $a_1 + \dots +a_k = 0$

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