Question

Degree of the field extension $K(x)\hookrightarrow\operatorname{Quot}\left(K[x,y]/(f)\right) $

score 103 · Answer 1 · 2015-12-20 11:52:06

103

Views

Degree of the field extension $K(x)\hookrightarrow\operatorname{Quot}\left(K[x,y]/(f)\right) $

Published on 20 Dec 2015 - 11:52

score 1.4k · Answer 2 · 2015-12-20 19:33:22

1.4k

Views

If $a,b\in K$ are algebraic over $k$, then $a\pm b, ab$ and $ab^{-1}$ are also algebraic.

Published on 20 Dec 2015 - 19:33

score 77 · Answer 3 · 2015-12-20 20:02:34

77

Views

Why if $K$ is a finite field, $|K|=p^d$ for a prime $p$?

Published on 20 Dec 2015 - 20:02

score 243 · Answer 4 · 2015-12-21 03:59:21

243

Views

f(x) is irreducible over K if and only if Gal(F/K) acts transitively on the roots of f(x).

Published on 21 Dec 2015 - 3:59

score 481 · Answer 5 · 2026-04-11 03:28:48

481

Views

Complex subfields of finite index

Published on 11 Apr 2026 - 3:28

score 97 · Answer 6 · 2015-12-23 20:29:05

97

Views

Show that $F_1\cap F_2/K$ is normal if $F_1/K$ and $F_2/K$ are normal.

Published on 23 Dec 2015 - 20:29

score 937 · Answer 7 · 2015-12-23 22:26:10

937

Views

If a field $F$ is an algebraic extension of a field $K$ then $(F:K)=(F(x):K(x))$

Published on 23 Dec 2015 - 22:26

score 268 · Answer 8 · 2015-12-23 22:59:45

268

Views

Show that if a field is perfect any irreducible polynomial is separable.

Published on 23 Dec 2015 - 22:59

score 1.8k · Answer 9 · 2026-03-28 03:27:48

1.8k

Views

Irreducible polynomial of degree $3$ and degree of extension

Published on 28 Mar 2026 - 3:27

score 882 · Answer 10 · 2026-04-06 07:13:09

882

Views

Irreducible polynomial over $\mathbb{Q}$ can not have repeated root in $\mathbb{C}$

Published on 06 Apr 2026 - 7:13

score 261 · Answer 11 · 2026-03-28 03:27:33

261

Views

Splitting field of $x^3-10x+10$ over $\mathbb{Q}(\sqrt{13})$

Published on 28 Mar 2026 - 3:27

score 635 · Answer 12 · 2015-12-27 16:15:03

635

Views

Maximal separable sub-extensions and Galois groups.

Published on 27 Dec 2015 - 16:15

score 95 · Answer 13 · 2026-04-12 08:42:08

95

Views

Examples where $H\ne \mathrm{Aut}(E/E^H)$

Published on 12 Apr 2026 - 8:42

score 387 · Answer 14 · 2026-04-12 23:29:16

387

Views

Tensor products over Ring of Integers

Published on 12 Apr 2026 - 23:29

score 65 · Answer 15 · 2026-03-28 03:25:24

65

Views

Splitting field is countable

Published on 28 Mar 2026 - 3:25

List Question

Degree of the field extension $K(x)\hookrightarrow\operatorname{Quot}\left(K[x,y]/(f)\right) $

If $a,b\in K$ are algebraic over $k$, then $a\pm b, ab$ and $ab^{-1}$ are also algebraic.

Why if $K$ is a finite field, $|K|=p^d$ for a prime $p$?

f(x) is irreducible over K if and only if Gal(F/K) acts transitively on the roots of f(x).

Complex subfields of finite index

Show that $F_1\cap F_2/K$ is normal if $F_1/K$ and $F_2/K$ are normal.

If a field $F$ is an algebraic extension of a field $K$ then $(F:K)=(F(x):K(x))$

Show that if a field is perfect any irreducible polynomial is separable.

Irreducible polynomial of degree $3$ and degree of extension

Irreducible polynomial over $\mathbb{Q}$ can not have repeated root in $\mathbb{C}$

Splitting field of $x^3-10x+10$ over $\mathbb{Q}(\sqrt{13})$

Maximal separable sub-extensions and Galois groups.

Examples where $H\ne \mathrm{Aut}(E/E^H)$

Tensor products over Ring of Integers

Splitting field is countable

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