Question

Why is $\Bbb Q(\sqrt[3]{2},\sqrt{-3})$ the splitting field of $x^3-2$?

score 65 · Answer 1 · 2015-02-14 22:14:23

65

Views

Why is $\Bbb Q(\sqrt[3]{2},\sqrt{-3})$ the splitting field of $x^3-2$?

Published on 14 Feb 2015 - 22:14

score 168 · Answer 2 · 2026-03-28 22:31:13

168

Views

Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$?

Published on 28 Mar 2026 - 22:31

score 376 · Answer 3 · 2026-04-07 04:57:10

376

Views

Showing that the fixed points of a homomorphism form a finite field.

Published on 07 Apr 2026 - 4:57

score 552 · Answer 4 · 2026-04-07 04:58:42

552

Views

$K$-monomorphism $L\rightarrow L$ is $K$-automorphism if finite extension $L:K$

Published on 07 Apr 2026 - 4:58

score 69 · Answer 5 · 2015-02-18 13:11:27

69

Views

$[F(a):F] \leq [\mathbb{Q}(a):\mathbb{Q}]$ - proof?

Published on 18 Feb 2015 - 13:11

score 827 · Answer 6 · 2015-02-19 07:12:50

827

Views

Field extensions and monomorphism

Published on 19 Feb 2015 - 7:12

score 356 · Answer 7 · 2015-02-20 04:11:13

356

Views

Writing elements of a finitely generated field extension in terms of the generators

Published on 20 Feb 2015 - 4:11

score 857 · Answer 8 · 2015-02-21 20:02:34

857

Views

Show that $\sqrt{2}\notin \mathbb{Q}(\sqrt[4]{3})$

Published on 21 Feb 2015 - 20:02

score 214 · Answer 9 · 2015-02-22 17:41:37

214

Views

Subring generated by $x$ is an integral domain iff it is a field iff the minimal polynomial of $x$ is irreducible

Published on 22 Feb 2015 - 17:41

score 230 · Answer 10 · 2015-02-23 01:25:47

230

Views

Let $K\subset L$ a finite field extension and $G=G(L/K)$ a Galois group of $L$ over $K$ then $|G|\leq[L:K]$.

Published on 23 Feb 2015 - 1:25

score 266 · Answer 11 · 2015-02-25 03:49:59

266

Views

Is there a field $K[\alpha]$ with $\alpha$ idempotent?

Published on 25 Feb 2015 - 3:49

score 555 · Answer 12 · 2015-02-25 16:00:37

555

Views

Why is the algebraic closure of $\Bbb{Q}$ not a finitely generated $\Bbb{Q}$-module?

Published on 25 Feb 2015 - 16:00

score 360 · Answer 13 · 2026-04-08 20:12:10

360

Views

The fundamental theorem of Galois theory

Published on 08 Apr 2026 - 20:12

score 123 · Answer 14 · 2015-02-26 17:46:07

123

Views

Necessary condition for the extension $F(\alpha) /F$ to be algebraic?

Published on 26 Feb 2015 - 17:46

score 98 · Answer 15 · 2026-04-07 08:05:53

98

Views

If $E,F$ are finite fields and $F\subseteq E,$ why is $E$ a finite-dimensional vector space over $F$?

Published on 07 Apr 2026 - 8:05

List Question

Why is $\Bbb Q(\sqrt[3]{2},\sqrt{-3})$ the splitting field of $x^3-2$?

Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$?

Showing that the fixed points of a homomorphism form a finite field.

$K$-monomorphism $L\rightarrow L$ is $K$-automorphism if finite extension $L:K$

$[F(a):F] \leq [\mathbb{Q}(a):\mathbb{Q}]$ - proof?

Field extensions and monomorphism

Writing elements of a finitely generated field extension in terms of the generators

Show that $\sqrt{2}\notin \mathbb{Q}(\sqrt[4]{3})$

Subring generated by $x$ is an integral domain iff it is a field iff the minimal polynomial of $x$ is irreducible

Let $K\subset L$ a finite field extension and $G=G(L/K)$ a Galois group of $L$ over $K$ then $|G|\leq[L:K]$.

Is there a field $K[\alpha]$ with $\alpha$ idempotent?

Why is the algebraic closure of $\Bbb{Q}$ not a finitely generated $\Bbb{Q}$-module?

The fundamental theorem of Galois theory

Necessary condition for the extension $F(\alpha) /F$ to be algebraic?

If $E,F$ are finite fields and $F\subseteq E,$ why is $E$ a finite-dimensional vector space over $F$?

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