Question

Elements of Galois group map roots of minimal polynomial to which other roots?

score 207 · Answer 1 · 2021-04-16 01:49:50

207

Views

Elements of Galois group map roots of minimal polynomial to which other roots?

Published on 16 Apr 2021 - 1:49

score 236 · Answer 2 · 2026-02-23 19:23:36

236

Views

Can inseparable elements “appear” in the residue field of the Galois closure of a field extension with separable residue field extension?

Published on 23 Feb 2026 - 19:23

score 203 · Answer 3 · 2021-04-22 07:10:12

203

Views

Suppose $K$ is a Galois extension of $F$. Consider $E/F$ a finite extension such that $K\cap E=F$. Show that $[KE:K]=[E:F]$.

Published on 22 Apr 2021 - 7:10

score 273 · Answer 4 · 2021-04-25 06:36:18

273

Views

Existence of a Field not containing $\sqrt{2}$, such that any finite extension is cyclic.

Published on 25 Apr 2021 - 6:36

score 91 · Answer 5 · 2021-04-25 11:27:03

91

Views

Galois group for $\operatorname{Gal}(\mathbb{Q}(\zeta_n + \zeta_n^{-1})/\mathbb{Q})$

Published on 25 Apr 2021 - 11:27

score 40 · Answer 6 · 2021-04-26 00:56:24

40

Views

A question related to abelian extensions

Published on 26 Apr 2021 - 0:56

score 328 · Answer 7 · 2021-04-26 17:00:03

328

Views

How can I prove that $\sqrt{2}$ is not inside Q($\sqrt{5 + \sqrt{5}}$)?

Published on 26 Apr 2021 - 17:00

score 111 · Answer 8 · 2021-04-28 10:49:04

111

Views

Criterion for finite Galois extension

Published on 28 Apr 2021 - 10:49

score 74 · Answer 9 · 2021-04-29 03:03:07

74

Views

What does it mean when one talks about splitting field of a multivariable polynomial? And then, Galois group of that splitting field?

Published on 29 Apr 2021 - 3:03

score 64 · Answer 10 · 2021-05-01 20:57:53

64

Views

Proving the irreducibility of a polynomial based on its Galois group

Published on 01 May 2021 - 20:57

score 215 · Answer 11 · 2021-05-02 14:38:11

215

Views

E/F is Galois ext with $[E:F]=p^2$ and its Galois group is not cyclic, then $\exists K$ a proper subfield of $E$.

Published on 02 May 2021 - 14:38

score 205 · Answer 12 · 2026-02-22 23:10:57

205

Views

Equivalent Definitions of Field Trace of Galois Extension

Published on 22 Feb 2026 - 23:10

score 163 · Answer 13 · 2021-05-09 23:13:16

163

Views

splitting field of separable and irreducible polynomial: does isomorphism of Galois subgroups imply isomorphism of subfields?

Published on 09 May 2021 - 23:13

score 168 · Answer 14 · 2021-05-10 17:16:21

168

Views

Splitting field of a degree 4 irreducible polynomial in $\mathbb{Q}[x]$ with two real roots and two non real roots

Published on 10 May 2021 - 17:16

score 60 · Answer 15 · 2021-05-12 02:30:35

60

Views

Degree of a polynomial zero over the simple extension attained by adjoining another zero.

Published on 12 May 2021 - 2:30

List Question

Elements of Galois group map roots of minimal polynomial to which other roots?

Can inseparable elements “appear” in the residue field of the Galois closure of a field extension with separable residue field extension?

Suppose $K$ is a Galois extension of $F$. Consider $E/F$ a finite extension such that $K\cap E=F$. Show that $[KE:K]=[E:F]$.

Existence of a Field not containing $\sqrt{2}$, such that any finite extension is cyclic.

Galois group for $\operatorname{Gal}(\mathbb{Q}(\zeta_n + \zeta_n^{-1})/\mathbb{Q})$

A question related to abelian extensions

How can I prove that $\sqrt{2}$ is not inside Q($\sqrt{5 + \sqrt{5}}$)?

Criterion for finite Galois extension

What does it mean when one talks about splitting field of a multivariable polynomial? And then, Galois group of that splitting field?

Proving the irreducibility of a polynomial based on its Galois group

E/F is Galois ext with $[E:F]=p^2$ and its Galois group is not cyclic, then $\exists K$ a proper subfield of $E$.

Equivalent Definitions of Field Trace of Galois Extension

splitting field of separable and irreducible polynomial: does isomorphism of Galois subgroups imply isomorphism of subfields?

Splitting field of a degree 4 irreducible polynomial in $\mathbb{Q}[x]$ with two real roots and two non real roots

Degree of a polynomial zero over the simple extension attained by adjoining another zero.

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