Question

If $\{a_{1},\ldots,a_{n}\}$ is a $K$-basis of $F$, then $\{1\otimes a_{1},\ldots,1\otimes a_{n}\}$ is an $E$-basis of $E\otimes_{K} F$.

score 114 · Answer 1 · 2026-04-01 00:53:28

114

Views

If $\{a_{1},\ldots,a_{n}\}$ is a $K$-basis of $F$, then $\{1\otimes a_{1},\ldots,1\otimes a_{n}\}$ is an $E$-basis of $E\otimes_{K} F$.

Published on 01 Apr 2026 - 0:53

score 604 · Answer 2 · 2026-03-23 23:31:58

604

Views

Is $\mathbb{Q}(\sqrt{2+\sqrt{-5}})$ normal over $\mathbb{Q}$

Published on 23 Mar 2026 - 23:31

score 487 · Answer 3 · 2026-04-05 23:30:01

487

Views

Comparing degrees of field extensions: compositum over field versus field over intersection.

Published on 05 Apr 2026 - 23:30

score 2.5k · Answer 4 · 2015-10-17 12:49:58

2.5k

Views

Let $E$ be an algebraic extension of $F$. If every polynomial in $F[x]$ splits in $E$, show that $E$ is algebraically closed.

Published on 17 Oct 2015 - 12:49

score 668 · Answer 5 · 2015-10-18 07:07:11

668

Views

Correspondence between Polynomial Roots and Automorphism Group

Published on 18 Oct 2015 - 7:07

score 70 · Answer 6 · 2015-10-18 09:50:14

70

Views

Equivalence of $Gal \left( \frac{F}{\Bbb{Q}} \right)$ to permutation group of polynomial

Published on 18 Oct 2015 - 9:50

score 128 · Answer 7 · 2026-04-01 00:50:51

128

Views

What is the minimal polynomial of $\alpha = \frac{3^{1/2}}{1+2^{1/3}} $ over $\mathbb{Q}$?

Published on 01 Apr 2026 - 0:50

score 1.2k · Answer 8 · 2015-10-21 21:25:57

1.2k

Views

Subfields of $\mathbb{C}$ which extend to $\mathbb{C}$ via finite extensions

Published on 21 Oct 2015 - 21:25

score 79 · Answer 9 · 2015-10-22 00:47:51

79

Views

Equal Field Extensions

Published on 22 Oct 2015 - 0:47

score 103 · Answer 10 · 2026-04-10 19:13:22

103

Views

$a,b$ algebraic over $F$, $([F(a):F], [F(b):F]) = 1 \implies m_b(X)$ irreducible in $F(a)[X]$.

Published on 10 Apr 2026 - 19:13

score 759 · Answer 11 · 2015-10-23 08:44:48

759

Views

Trace of an element of a number field is in $\mathbb{Q}$. A strange proof

Published on 23 Oct 2015 - 8:44

score 4k · Answer 12 · 2026-03-03 08:35:59

4k

Views

Minimal polynomial of $\sqrt{2}+\sqrt{3}$ over $\mathbb Q$

Published on 03 Mar 2026 - 8:35

score 83 · Answer 13 · 2015-10-23 20:05:07

83

Views

$K\subseteq L\subseteq M$ be fields. If $L/K$ and $M/L$ are both separable then $M/K$ is separable

Published on 23 Oct 2015 - 20:05

score 943 · Answer 14 · 2015-10-25 17:14:44

943

Views

Adjoining a complex number to $\mathbb{Q}$

Published on 25 Oct 2015 - 17:14

score 60 · Answer 15 · 2015-10-26 18:02:37

60

Views

Why does the definition of a simple extension make use of polynomial fractions?

Published on 26 Oct 2015 - 18:02

List Question

If $\{a_{1},\ldots,a_{n}\}$ is a $K$-basis of $F$, then $\{1\otimes a_{1},\ldots,1\otimes a_{n}\}$ is an $E$-basis of $E\otimes_{K} F$.

Is $\mathbb{Q}(\sqrt{2+\sqrt{-5}})$ normal over $\mathbb{Q}$

Comparing degrees of field extensions: compositum over field versus field over intersection.

Let $E$ be an algebraic extension of $F$. If every polynomial in $F[x]$ splits in $E$, show that $E$ is algebraically closed.

Correspondence between Polynomial Roots and Automorphism Group

Equivalence of $Gal \left( \frac{F}{\Bbb{Q}} \right)$ to permutation group of polynomial

What is the minimal polynomial of $\alpha = \frac{3^{1/2}}{1+2^{1/3}} $ over $\mathbb{Q}$?

Subfields of $\mathbb{C}$ which extend to $\mathbb{C}$ via finite extensions

Equal Field Extensions

$a,b$ algebraic over $F$, $([F(a):F], [F(b):F]) = 1 \implies m_b(X)$ irreducible in $F(a)[X]$.

Trace of an element of a number field is in $\mathbb{Q}$. A strange proof

Minimal polynomial of $\sqrt{2}+\sqrt{3}$ over $\mathbb Q$

$K\subseteq L\subseteq M$ be fields. If $L/K$ and $M/L$ are both separable then $M/K$ is separable

Adjoining a complex number to $\mathbb{Q}$

Why does the definition of a simple extension make use of polynomial fractions?

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