Question

Let $F $ be a field with char $F =p >0$. If $\alpha \in K/F$ is separable over $F $, prove that $F (\alpha)/F $ is separable.

score 152 · Answer 1 · 2026-03-22 16:35:08

152

Views

Let $F $ be a field with char $F =p >0$. If $\alpha \in K/F$ is separable over $F $, prove that $F (\alpha)/F $ is separable.

Published on 22 Mar 2026 - 16:35

score 31 · Answer 2 · 2018-05-17 04:58:47

31

Views

Show that E is a finite extension of F

Published on 17 May 2018 - 4:58

score 102 · Answer 3 · 2018-05-17 07:05:25

102

Views

On using Steinitz Primitive Theorem

Published on 17 May 2018 - 7:05

score 596 · Answer 4 · 2018-05-17 07:13:00

596

Views

Find the splitting field of this polynomial over GF(2)

Published on 17 May 2018 - 7:13

score 370 · Answer 5 · 2026-03-25 21:47:54

370

Views

Inverse element in field extension $\mathbb{Q}(\sqrt{2},\sqrt{3})$

Published on 25 Mar 2026 - 21:47

score 342 · Answer 6 · 2026-03-25 21:51:15

342

Views

Let $w=e^{2\pi i/7}$. Find the lattice of subgroups of Gal$(\mathbb Q(w)/\mathbb Q)$ and of intermediate fields in $\mathbb Q(w)\supseteq\mathbb Q$

Published on 25 Mar 2026 - 21:51

score 67 · Answer 7 · 2026-03-26 08:03:26

67

Views

A reducibility of $x^{14}+1$ in the field

Published on 26 Mar 2026 - 8:03

score 72 · Answer 8 · 2026-03-26 03:18:34

72

Views

Solve the equation in the field

Published on 26 Mar 2026 - 3:18

score 1k · Answer 9 · 2026-03-26 06:21:32

1k

Views

Is $x^4+1$ irreducible in $F_3[x]$?

Published on 26 Mar 2026 - 6:21

score 727 · Answer 10 · 2026-03-25 21:54:14

727

Views

$K/L$ is normal and $L/F$ is purely inseparable. Show that $K/F$ is normal (Proof verification)

Published on 25 Mar 2026 - 21:54

score 528 · Answer 11 · 2026-03-25 21:54:05

528

Views

Let $a \in F - F^{p}$. Show that $x^{p} - a$ is irreducible over $F$.

Published on 25 Mar 2026 - 21:54

score 161 · Answer 12 · 2018-05-17 23:27:46

161

Views

Is $(x,y) \mapsto (\frac xy, x^2-y^3)$ an automorphism of $k(x,y)$?

Published on 17 May 2018 - 23:27

score 145 · Answer 13 · 2026-03-25 23:13:03

145

Views

Let $F = F_0(t_1^p, t_2^p)$ and $L = F_0(t_1, t_2)$. Show that if $\theta \in L - F$ then $[F(\theta) : F] = p$

Published on 25 Mar 2026 - 23:13

score 144 · Answer 14 · 2026-03-27 23:48:09

144

Views

finding $[K(α):K] $, and $\operatorname{Aut}(K(α)/K)$ when $K=F_3(t)$ and $α$ is root of the polynomial $f(x)=x^6−tx^3+1$

Published on 27 Mar 2026 - 23:48

score 190 · Answer 15 · 2026-03-26 06:22:31

190

Views

The degrees of irreducible divisors of polynomials

Published on 26 Mar 2026 - 6:22

List Question

Let $F $ be a field with char $F =p >0$. If $\alpha \in K/F$ is separable over $F $, prove that $F (\alpha)/F $ is separable.

Show that E is a finite extension of F

On using Steinitz Primitive Theorem

Find the splitting field of this polynomial over GF(2)

Inverse element in field extension $\mathbb{Q}(\sqrt{2},\sqrt{3})$

Let $w=e^{2\pi i/7}$. Find the lattice of subgroups of Gal$(\mathbb Q(w)/\mathbb Q)$ and of intermediate fields in $\mathbb Q(w)\supseteq\mathbb Q$

A reducibility of $x^{14}+1$ in the field

Solve the equation in the field

Is $x^4+1$ irreducible in $F_3[x]$?

$K/L$ is normal and $L/F$ is purely inseparable. Show that $K/F$ is normal (Proof verification)

Let $a \in F - F^{p}$. Show that $x^{p} - a$ is irreducible over $F$.

Is $(x,y) \mapsto (\frac xy, x^2-y^3)$ an automorphism of $k(x,y)$?

Let $F = F_0(t_1^p, t_2^p)$ and $L = F_0(t_1, t_2)$. Show that if $\theta \in L - F$ then $[F(\theta) : F] = p$

finding $[K(α):K] $, and $\operatorname{Aut}(K(α)/K)$ when $K=F_3(t)$ and $α$ is root of the polynomial $f(x)=x^6−tx^3+1$

The degrees of irreducible divisors of polynomials

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