Question

Find Galois Group of $4x^4+5x^3-9$ over $\mathbb{Q}$

score 87 · Answer 1 · 2017-08-31 16:02:54

87

Views

Find Galois Group of $4x^4+5x^3-9$ over $\mathbb{Q}$

Published on 31 Aug 2017 - 16:02

score 54 · Answer 2 · 2017-09-01 13:28:34

54

Views

about the Galois extension

Published on 01 Sep 2017 - 13:28

score 97 · Answer 3 · 2017-09-13 09:29:48

97

Views

Splitting field of $X^4+X^3+1\in\mathbb{Q}[X]$

Published on 13 Sep 2017 - 9:29

score 556 · Answer 4 · 2017-09-18 17:47:03

556

Views

Determine whether the following fields are Galois over $\mathbb{Q}$.

Published on 18 Sep 2017 - 17:47

score 2.2k · Answer 5 · 2017-09-18 18:43:52

2.2k

Views

If $F\subseteq L\subseteq K$ are fields with $K/L$ and $L/F$ Galois, then $K/F$ is Galois?.

Published on 18 Sep 2017 - 18:43

score 685 · Answer 6 · 2017-09-19 00:22:36

685

Views

Let $k$ be a field, and let $K = k(x)$ be the rational function field in one variable over $k$. Let $\sigma$ and $\tau$ be the automorphisms of $K$

Published on 19 Sep 2017 - 0:22

score 88 · Answer 7 · 2017-09-20 03:13:22

88

Views

Give an example of fields $k\subseteq K\subseteq L $, and $l\subseteq L$, for which $l/k$ and $L/K$ are algebraic, $k$ is algebraically closed

Published on 20 Sep 2017 - 3:13

score 1.9k · Answer 8 · 2017-10-16 15:48:55

1.9k

Views

Let $K$ and $L$ be extensions of $F$. Show that $KL$ is Galois over $F$ if both $K$ and $L$ are Galois over $F$. Is the converse true?

Published on 16 Oct 2017 - 15:48

score 42 · Answer 9 · 2026-03-26 04:32:07

42

Views

Showing that a certain residue field is finite

Published on 26 Mar 2026 - 4:32

score 277 · Answer 10 · 2026-03-26 16:07:10

277

Views

$L/K$ unramified at $v$ and $\widetilde{K}$ intermediate field $\Rightarrow\, \widetilde{K}/K$ unramified at $v$

Published on 26 Mar 2026 - 16:07

score 55 · Answer 11 · 2026-03-26 22:57:56

55

Views

$\mathbb{Z}/4$-Galois cover of $\mathbb{Q}_p$

Published on 26 Mar 2026 - 22:57

score 85 · Answer 12 · 2017-11-12 02:51:46

85

Views

In the following problems, let $K$ be the splitting field of $f(x)$ over $F$. Determine Gal$(K/ F)$ and find all the intermediate subfields of $K/F$.

Published on 12 Nov 2017 - 2:51

score 288 · Answer 13 · 2017-11-12 21:46:31

288

Views

Let $K$ be a Galois extension of $\mathbb{Q}$. View $K$ as a subfield of $\mathbb{C}$. If $\sigma$ is complex conjugation, show that $\sigma(K) = K$

Published on 12 Nov 2017 - 21:46

score 1.2k · Answer 14 · 2017-11-12 23:14:22

1.2k

Views

Show that $Q_8$ is not isomorphic to a subgroup of $S_4$. Conclude that $Q_8$ is not the Galois group of a degree $4$ polynomial.

Published on 12 Nov 2017 - 23:14

score 57 · Answer 15 · 2017-11-19 18:10:58

57

Views

$Gal(F/k) \cong \langle k^{^{n}}, a_1, \ldots, a_n \rangle / \langle k^{^{n}} \rangle$

Published on 19 Nov 2017 - 18:10

List Question

Find Galois Group of $4x^4+5x^3-9$ over $\mathbb{Q}$

about the Galois extension

Splitting field of $X^4+X^3+1\in\mathbb{Q}[X]$

Determine whether the following fields are Galois over $\mathbb{Q}$.

If $F\subseteq L\subseteq K$ are fields with $K/L$ and $L/F$ Galois, then $K/F$ is Galois?.

Let $k$ be a field, and let $K = k(x)$ be the rational function field in one variable over $k$. Let $\sigma$ and $\tau$ be the automorphisms of $K$

Give an example of fields $k\subseteq K\subseteq L $, and $l\subseteq L$, for which $l/k$ and $L/K$ are algebraic, $k$ is algebraically closed

Let $K$ and $L$ be extensions of $F$. Show that $KL$ is Galois over $F$ if both $K$ and $L$ are Galois over $F$. Is the converse true?

Showing that a certain residue field is finite

$L/K$ unramified at $v$ and $\widetilde{K}$ intermediate field $\Rightarrow\, \widetilde{K}/K$ unramified at $v$

$\mathbb{Z}/4$-Galois cover of $\mathbb{Q}_p$

In the following problems, let $K$ be the splitting field of $f(x)$ over $F$. Determine Gal$(K/ F)$ and find all the intermediate subfields of $K/F$.

Let $K$ be a Galois extension of $\mathbb{Q}$. View $K$ as a subfield of $\mathbb{C}$. If $\sigma$ is complex conjugation, show that $\sigma(K) = K$

Show that $Q_8$ is not isomorphic to a subgroup of $S_4$. Conclude that $Q_8$ is not the Galois group of a degree $4$ polynomial.

$Gal(F/k) \cong \langle k^{^{n}}, a_1, \ldots, a_n \rangle / \langle k^{^{n}} \rangle$

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