Question

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

score 90 · Answer 1 · 2026-04-15 18:03:48

90

Views

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

Published on 15 Apr 2026 - 18:03

score 57 · Answer 2 · 2026-04-10 01:52:56

57

Views

about the Galois extension

Published on 10 Apr 2026 - 1:52

score 101 · Answer 3 · 2026-04-20 07:48:23

101

Views

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

Published on 20 Apr 2026 - 7:48

score 559 · Answer 4 · 2026-04-16 05:01:07

559

Views

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

Published on 16 Apr 2026 - 5:01

score 2.2k · Answer 5 · 2026-04-16 14:21:08

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 16 Apr 2026 - 14:21

score 689 · Answer 6 · 2026-04-14 18:23:17

689

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 14 Apr 2026 - 18:23

score 92 · Answer 7 · 2026-04-14 07:09:45

92

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 14 Apr 2026 - 7:09

score 1.9k · Answer 8 · 2026-04-16 06:30:09

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 Apr 2026 - 6:30

score 44 · Answer 9 · 2026-04-16 08:13:42

44

Views

Showing that a certain residue field is finite

Published on 16 Apr 2026 - 8:13

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 87 · Answer 12 · 2026-04-15 18:58:49

87

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 15 Apr 2026 - 18:58

score 291 · Answer 13 · 2026-04-15 02:53:29

291

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 15 Apr 2026 - 2:53

score 1.2k · Answer 14 · 2026-04-15 04:15:53

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 15 Apr 2026 - 4:15

score 60 · Answer 15 · 2026-04-15 18:13:19

60

Views

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

Published on 15 Apr 2026 - 18:13

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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