Question

Why is $\text{Gal}(K/\mathbb{Q}) \cong G_{\mathbb{Q}}/{\{\sigma \in G_{\mathbb{Q}}: \ \sigma|_K=id_K \}}$?

score 120 · Answer 1 · 2026-03-29 04:11:07

120

Views

Why is $\text{Gal}(K/\mathbb{Q}) \cong G_{\mathbb{Q}}/{\{\sigma \in G_{\mathbb{Q}}: \ \sigma|_K=id_K \}}$?

Published on 29 Mar 2026 - 4:11

score 420 · Answer 2 · 2026-03-29 05:50:06

420

Views

For a complex number $\alpha $ which is algebraic over $\Bbb Q$, determining whether $\bar{\alpha}\in \Bbb Q(\alpha)$ or not

Published on 29 Mar 2026 - 5:50

score 690 · Answer 3 · 2026-03-29 04:10:51

690

Views

Determine the Galois group of $x^3 + 3x^2 - 1$ over $\mathbb{Q}$

Published on 29 Mar 2026 - 4:10

score 861 · Answer 4 · 2026-03-29 05:43:30

861

Views

If Gal$(K/\mathbb{Q}) = S_5$ then $K$ is the splitting field of a degree $5$ polynomial

Published on 29 Mar 2026 - 5:43

score 315 · Answer 5 · 2026-03-29 05:44:15

315

Views

Let $f(x) \in \mathbb{Q}[x]$ be an irreducible polynomial of degree $4$ with exactly $2$ real roots.

Published on 29 Mar 2026 - 5:44

score 112 · Answer 6 · 2026-04-17 07:27:53

112

Views

Let $E \supset F \supset G$ and $G = E^{\rm{Aut}(E/G)}$. Then is it true that $F = E^{\rm{Aut}(E/F)}$?

Published on 17 Apr 2026 - 7:27

score 92 · Answer 7 · 2026-04-12 13:36:48

92

Views

Show that $(L : K), (Z : K)$ and $(L : Z)$ are all Galois extensions, but that $Z$ is not a stable intermediate field of $(L : K)$.

Published on 12 Apr 2026 - 13:36

score 214 · Answer 8 · 2026-04-16 06:25:25

214

Views

$(K(x) : F(U))$ is a Galois extension then $U$ is finite.

Published on 16 Apr 2026 - 6:25

score 64 · Answer 9 · 2026-04-16 11:35:43

64

Views

Show that $U = \langle\sigma\rangle$ is an infinite cyclic group; determine its fixed field $F(U)$ and the degree $[K(x) : F(U)]$.

Published on 16 Apr 2026 - 11:35

score 107 · Answer 10 · 2026-04-16 06:33:17

107

Views

Determine whether or not there is an automorphism $\sigma \in Gal(M/F)$ with $\sigma (a) = a'$ and $\sigma(b) = b'$.

Published on 16 Apr 2026 - 6:33

score 202 · Answer 11 · 2026-04-14 01:36:01

202

Views

$k$ is algebraically closed in $K$, and $lK=L$ , but $l$ is not algebraically closed in $L$.

Published on 14 Apr 2026 - 1:36

score 271 · Answer 12 · 2026-04-16 21:45:15

271

Views

$[K : F]_s = [K : L]_s [L : F]_s $ and $[K : F]_i = [K : L]_i [L : F]_i $

Published on 16 Apr 2026 - 21:45

score 72 · Answer 13 · 2026-04-12 20:34:48

72

Views

$Z = \{x \in K \mid \sigma_1(x) = \cdots = \sigma_n(x)\}$ is a subfield of K with $[K : Z] \geq n$.

Published on 12 Apr 2026 - 20:34

score 202 · Answer 14 · 2026-05-13 20:17:20

202

Views

degree of minimal polynomial of $\alpha$ is same as degree of minimal polynomial of $\sigma(\alpha)$

Published on 13 May 2026 - 20:17

score 1.1k · Answer 15 · 2026-04-15 13:08:56

1.1k

Views

$E$ is a splitting field of a polynomial over $K$ then $E/K$ is finite normal extension

Published on 15 Apr 2026 - 13:08

List Question

Why is $\text{Gal}(K/\mathbb{Q}) \cong G_{\mathbb{Q}}/{\{\sigma \in G_{\mathbb{Q}}: \ \sigma|_K=id_K \}}$?

For a complex number $\alpha $ which is algebraic over $\Bbb Q$, determining whether $\bar{\alpha}\in \Bbb Q(\alpha)$ or not

Determine the Galois group of $x^3 + 3x^2 - 1$ over $\mathbb{Q}$

If Gal$(K/\mathbb{Q}) = S_5$ then $K$ is the splitting field of a degree $5$ polynomial

Let $f(x) \in \mathbb{Q}[x]$ be an irreducible polynomial of degree $4$ with exactly $2$ real roots.

Let $E \supset F \supset G$ and $G = E^{\rm{Aut}(E/G)}$. Then is it true that $F = E^{\rm{Aut}(E/F)}$?

Show that $(L : K), (Z : K)$ and $(L : Z)$ are all Galois extensions, but that $Z$ is not a stable intermediate field of $(L : K)$.

$(K(x) : F(U))$ is a Galois extension then $U$ is finite.

Show that $U = \langle\sigma\rangle$ is an infinite cyclic group; determine its fixed field $F(U)$ and the degree $[K(x) : F(U)]$.

Determine whether or not there is an automorphism $\sigma \in Gal(M/F)$ with $\sigma (a) = a'$ and $\sigma(b) = b'$.

$k$ is algebraically closed in $K$, and $lK=L$ , but $l$ is not algebraically closed in $L$.

$[K : F]_s = [K : L]_s [L : F]_s $ and $[K : F]_i = [K : L]_i [L : F]_i $

$Z = \{x \in K \mid \sigma_1(x) = \cdots = \sigma_n(x)\}$ is a subfield of K with $[K : Z] \geq n$.

degree of minimal polynomial of $\alpha$ is same as degree of minimal polynomial of $\sigma(\alpha)$

$E$ is a splitting field of a polynomial over $K$ then $E/K$ is finite normal extension

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