Question

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

score 119 · Answer 1 · 2020-07-16 17:40:50

119

Views

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

Published on 16 Jul 2020 - 17:40

score 419 · Answer 2 · 2020-07-17 01:35:58

419

Views

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

Published on 17 Jul 2020 - 1:35

score 689 · Answer 3 · 2020-07-18 21:29:52

689

Views

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

Published on 18 Jul 2020 - 21:29

score 860 · Answer 4 · 2020-07-19 19:27:40

860

Views

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

Published on 19 Jul 2020 - 19:27

score 314 · Answer 5 · 2020-07-20 18:46:32

314

Views

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

Published on 20 Jul 2020 - 18:46

score 109 · Answer 6 · 2016-10-23 02:27:59

109

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 23 Oct 2016 - 2:27

score 89 · Answer 7 · 2017-01-09 03:29:35

89

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 09 Jan 2017 - 3:29

score 211 · Answer 8 · 2017-01-09 07:40:21

211

Views

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

Published on 09 Jan 2017 - 7:40

score 61 · Answer 9 · 2017-01-10 00:34:33

61

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 10 Jan 2017 - 0:34

score 104 · Answer 10 · 2017-01-25 19:16:58

104

Views

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

Published on 25 Jan 2017 - 19:16

score 199 · Answer 11 · 2017-01-26 01:40:25

199

Views

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

Published on 26 Jan 2017 - 1:40

score 268 · Answer 12 · 2017-01-26 05:20:23

268

Views

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

Published on 26 Jan 2017 - 5:20

score 69 · Answer 13 · 2017-01-27 13:53:37

69

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 27 Jan 2017 - 13:53

score 198 · Answer 14 · 2017-02-01 02:08:12

198

Views

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

Published on 01 Feb 2017 - 2:08

score 1.1k · Answer 15 · 2017-02-05 01:53:38

1.1k

Views

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

Published on 05 Feb 2017 - 1:53

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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