Question

$f,g \in \mathbb Q[x]$ , $f(a)=g(b)=0$ ; $f$ is irreducible in $\mathbb Q(b)[x]$ iff $g$ is irreducible in $\mathbb Q(a)[x]$?

score 99 · Answer 1 · 2026-04-17 14:40:26

99

Views

$f,g \in \mathbb Q[x]$ , $f(a)=g(b)=0$ ; $f$ is irreducible in $\mathbb Q(b)[x]$ iff $g$ is irreducible in $\mathbb Q(a)[x]$?

Published on 17 Apr 2026 - 14:40

score 278 · Answer 2 · 2026-04-12 03:31:38

278

Views

relationship between monomorphism and automorphism

Published on 12 Apr 2026 - 3:31

score 129 · Answer 3 · 2026-04-13 09:13:36

129

Views

showing that there are $n$ distinct $K$-monomorphisms

Published on 13 Apr 2026 - 9:13

score 2.2k · Answer 4 · 2026-04-16 06:28:13

2.2k

Views

Let $F$ be a field, and $K$ a field extension of $F$. Prove that $[K:F] = 1$ iff $K=F$.

Published on 16 Apr 2026 - 6:28

score 46 · Answer 5 · 2026-04-15 23:18:52

46

Views

Field automorphisms of extension generated by two coprime algebraic elements.

Published on 15 Apr 2026 - 23:18

score 2.5k · Answer 6 · 2026-04-16 19:15:43

2.5k

Views

splitting field of $x^n-1$ over $\mathbb{Q}$

Published on 16 Apr 2026 - 19:15

score 739 · Answer 7 · 2026-04-08 13:49:57

739

Views

Different number field discriminants in Sage and Magma

Published on 08 Apr 2026 - 13:49

score 50 · Answer 8 · 2026-04-13 04:58:46

50

Views

If every polynomial in $F[x]$ splits then there exists no nontrivial algebraic extension

Published on 13 Apr 2026 - 4:58

score 1.2k · Answer 9 · 2026-03-27 22:03:06

1.2k

Views

Example of finite field extension where root not separable

Published on 27 Mar 2026 - 22:03

score 68 · Answer 10 · 2026-04-17 04:05:41

68

Views

If $K \subset \mathbb{Q}(\sqrt[p]{2},\sqrt[q]{2})$ and $[K : \mathbb{Q}]=p$ then $K= \mathbb{Q}(\sqrt[p]{2})$

Published on 17 Apr 2026 - 4:05

score 122 · Answer 11 · 2026-04-15 10:29:18

122

Views

Proving an element belongs to field extension

Published on 15 Apr 2026 - 10:29

score 68 · Answer 12 · 2026-04-16 14:49:13

68

Views

Let $F$ be an extension field over $K$ , if $[F(x):K(x)]$ is finite , then is $F$ also a finite extension over $K$ ?

Published on 16 Apr 2026 - 14:49

score 60 · Answer 13 · 2026-04-11 22:07:49

60

Views

Degree of field extensions

Published on 11 Apr 2026 - 22:07

score 233 · Answer 14 · 2026-03-29 19:14:34

233

Views

Find $ [\mathbb{Q(\alpha)}: \mathbb{Q}] $ where $ \alpha=\zeta+\zeta^3+\zeta^4+\zeta^5+\zeta^9 $

Published on 29 Mar 2026 - 19:14

score 909 · Answer 15 · 2026-04-11 07:17:33

909

Views

A normal closure of an arbitrary field extension

Published on 11 Apr 2026 - 7:17

List Question

$f,g \in \mathbb Q[x]$ , $f(a)=g(b)=0$ ; $f$ is irreducible in $\mathbb Q(b)[x]$ iff $g$ is irreducible in $\mathbb Q(a)[x]$?

relationship between monomorphism and automorphism

showing that there are $n$ distinct $K$-monomorphisms

Let $F$ be a field, and $K$ a field extension of $F$. Prove that $[K:F] = 1$ iff $K=F$.

Field automorphisms of extension generated by two coprime algebraic elements.

splitting field of $x^n-1$ over $\mathbb{Q}$

Different number field discriminants in Sage and Magma

If every polynomial in $F[x]$ splits then there exists no nontrivial algebraic extension

Example of finite field extension where root not separable

If $K \subset \mathbb{Q}(\sqrt[p]{2},\sqrt[q]{2})$ and $[K : \mathbb{Q}]=p$ then $K= \mathbb{Q}(\sqrt[p]{2})$

Proving an element belongs to field extension

Let $F$ be an extension field over $K$ , if $[F(x):K(x)]$ is finite , then is $F$ also a finite extension over $K$ ?

Degree of field extensions

Find $ [\mathbb{Q(\alpha)}: \mathbb{Q}] $ where $ \alpha=\zeta+\zeta^3+\zeta^4+\zeta^5+\zeta^9 $

A normal closure of an arbitrary field extension

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