Question

Is there any irreducible polynomial over $\mathbb{Q}$ whose Galois group is $S_4$?

score 974 · Answer 1 · 2014-11-05 11:12:03

974

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Is there any irreducible polynomial over $\mathbb{Q}$ whose Galois group is $S_4$?

Published on 05 Nov 2014 - 11:12

score 33 · Answer 2 · 2014-11-05 19:55:16

33

Views

Let $G/H$ be a Galois extension and let $N_1$ and $N_2$ be subfields between $G$ and $H$

Published on 05 Nov 2014 - 19:55

score 55 · Answer 3 · 2026-03-26 04:36:22

55

Views

Find a positive integer $n$ such that there is a subfield of $\mathbb{Q}_n$ that is not a cyclotomic extension of $\mathbb{Q}$

Published on 26 Mar 2026 - 4:36

score 95 · Answer 4 · 2026-03-26 03:01:31

95

Views

Is there a direct proof that pi is not the root of an algebraic equation whose degree is a power of 2

Published on 26 Mar 2026 - 3:01

score 131 · Answer 5 · 2014-11-07 11:28:43

131

Views

P. Morandi on p-closure of a field

Published on 07 Nov 2014 - 11:28

score 7.6k · Answer 6 · 2026-04-01 07:17:20

7.6k

Views

Why can we prove mathematically that a formula to solve an (n+5) order polynomial does not exist?

Published on 01 Apr 2026 - 7:17

score 1k · Answer 7 · 2026-02-23 01:08:25

1k

Views

If $F/k$ normal, $g,h \in F[X]$ monic irreducible factors of $f \in k[X]$, then $\exists\ \sigma \in Aut(F/k)$ s.t. $g^\sigma = h$.

Published on 23 Feb 2026 - 1:08

score 722 · Answer 8 · 2026-03-25 11:04:48

722

Views

The absolute Galois group of a finite field is strongly complete

Published on 25 Mar 2026 - 11:04

score 1.6k · Answer 9 · 2026-04-02 19:12:59

1.6k

Views

Calculating automorphism groups

Published on 02 Apr 2026 - 19:12

score 80 · Answer 10 · 2014-11-10 22:01:13

80

Views

Galois Group of a rational polynomial

Published on 10 Nov 2014 - 22:01

score 3.8k · Answer 11 · 2026-04-02 19:13:29

3.8k

Views

Is Q(4th root(2))/Q a galois extension

Published on 02 Apr 2026 - 19:13

score 1.3k · Answer 12 · 2014-11-11 21:04:58

1.3k

Views

Are most rational quintics unsolvable?

Published on 11 Nov 2014 - 21:04

score 536 · Answer 13 · 2026-04-02 21:18:47

536

Views

Order of automorphism = Order of field extension property

Published on 02 Apr 2026 - 21:18

score 69 · Answer 14 · 2014-11-12 01:11:10

69

Views

Order of conjugate map

Published on 12 Nov 2014 - 1:11

score 2.7k · Answer 15 · 2026-04-02 21:15:23

2.7k

Views

Degree 4 extension of $\mathbb {Q}$ with no intermediate field

Published on 02 Apr 2026 - 21:15

List Question

Is there any irreducible polynomial over $\mathbb{Q}$ whose Galois group is $S_4$?

Let $G/H$ be a Galois extension and let $N_1$ and $N_2$ be subfields between $G$ and $H$

Find a positive integer $n$ such that there is a subfield of $\mathbb{Q}_n$ that is not a cyclotomic extension of $\mathbb{Q}$

Is there a direct proof that pi is not the root of an algebraic equation whose degree is a power of 2

P. Morandi on p-closure of a field

Why can we prove mathematically that a formula to solve an (n+5) order polynomial does not exist?

If $F/k$ normal, $g,h \in F[X]$ monic irreducible factors of $f \in k[X]$, then $\exists\ \sigma \in Aut(F/k)$ s.t. $g^\sigma = h$.

The absolute Galois group of a finite field is strongly complete

Calculating automorphism groups

Galois Group of a rational polynomial

Is Q(4th root(2))/Q a galois extension

Are most rational quintics unsolvable?

Order of automorphism = Order of field extension property

Order of conjugate map

Degree 4 extension of $\mathbb {Q}$ with no intermediate field

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