Question

How to show that the image of the morphism $X \mapsto X^{p^n}$ equals the separable closure.

score 72 · Answer 1 · 2023-03-11 16:09:53

72

Views

How to show that the image of the morphism $X \mapsto X^{p^n}$ equals the separable closure.

Published on 11 Mar 2023 - 16:09

score 107 · Answer 2 · 2023-03-14 08:27:25

107

Views

Separable extensions and simple roots

Published on 14 Mar 2023 - 8:27

score 57 · Answer 3 · 2023-03-17 12:03:58

57

Views

Minimum polynomial and inseparable degree

Published on 17 Mar 2023 - 12:03

score 92 · Answer 4 · 2023-05-09 23:35:23

92

Views

If $p:=x^n-x-a$ has a root in $K$, then $p$ splits into linear factors in $K[X]$

Published on 09 May 2023 - 23:35

score 42 · Answer 5 · 2023-05-10 06:32:06

42

Views

Let $L/K$ be finite extension, is $L/K_s$ purley inseperable

Published on 10 May 2023 - 6:32

score 121 · Answer 6 · 2023-05-11 22:15:26

121

Views

Example of a field on which every irreducible polynomial has degree a power of $p$

Published on 11 May 2023 - 22:15

score 48 · Answer 7 · 2023-05-12 06:45:21

48

Views

Field extension $L/K$ such that every element has degree $1$ or $2$ over $K$

Published on 12 May 2023 - 6:45

score 70 · Answer 8 · 2023-05-14 21:36:14

70

Views

If $L/K$ is normal extension $\Rightarrow K_s/K$ is normal extension

Published on 14 May 2023 - 21:36

score 62 · Answer 9 · 2023-05-16 19:52:08

62

Views

Separable extension characterization

Published on 16 May 2023 - 19:52

score 62 · Answer 10 · 2023-05-22 15:24:26

62

Views

Are all algebraic extensions of finite fields separable? What about fields of characteristic p in general?

Published on 22 May 2023 - 15:24

score 25 · Answer 11 · 2023-06-01 23:18:33

25

Views

If $I=F$ , $K/S$ purely inseparable, $S/F$ separable then $K/I$ is not separable

Published on 01 Jun 2023 - 23:18

score 82 · Answer 12 · 2023-06-18 07:38:11

82

Views

Is $x^{100} - x^2 + 1$ separable in an algebraic closure of $\mathbb{F}_2$

Published on 18 Jun 2023 - 7:38

score 41 · Answer 13 · 2023-06-29 07:34:02

41

Views

Classification of separable rings

Published on 29 Jun 2023 - 7:34

score 22 · Answer 14 · 2023-07-03 14:17:14

22

Views

prove $\deg_{\alpha,L}\mid\deg_{\alpha,K}$ if $\alpha$ is separable and $K(\alpha)/K$ is normal

Published on 03 Jul 2023 - 14:17

score 63 · Answer 15 · 2023-07-05 09:26:48

63

Views

Proof that on infinite fields, finite separable extensions are all simple.

Published on 05 Jul 2023 - 9:26

List Question

How to show that the image of the morphism $X \mapsto X^{p^n}$ equals the separable closure.

Separable extensions and simple roots

Minimum polynomial and inseparable degree

If $p:=x^n-x-a$ has a root in $K$, then $p$ splits into linear factors in $K[X]$

Let $L/K$ be finite extension, is $L/K_s$ purley inseperable

Example of a field on which every irreducible polynomial has degree a power of $p$

Field extension $L/K$ such that every element has degree $1$ or $2$ over $K$

If $L/K$ is normal extension $\Rightarrow K_s/K$ is normal extension

Separable extension characterization

Are all algebraic extensions of finite fields separable? What about fields of characteristic p in general?

If $I=F$ , $K/S$ purely inseparable, $S/F$ separable then $K/I$ is not separable

Is $x^{100} - x^2 + 1$ separable in an algebraic closure of $\mathbb{F}_2$

Classification of separable rings

prove $\deg_{\alpha,L}\mid\deg_{\alpha,K}$ if $\alpha$ is separable and $K(\alpha)/K$ is normal

Proof that on infinite fields, finite separable extensions are all simple.

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