List questions from field-theory

115

Views

Is there a polynomial which detects when the two smallest roots of a given real polynomial are equal?

Published on 25 Mar 2026 - 16:02

90

Views

How to show that if $\pi(x) \mid x^{p^n} - x$ then $\deg(\pi(x)) \mid n$?

Published on 15 Apr 2026 - 10:48

56

Views

Finding $f= g(s_1,s_2,s_3)$ if $char(K) = p$

Published on 17 Apr 2026 - 6:30

65

Views

Cartesian product of a field as a ring extension

Published on 13 Apr 2026 - 5:17

176

Views

Finding splitting fields over $\mathbb{Q}(\sqrt{-3})$

Published on 14 Apr 2026 - 4:46

172

Views

Intersection of Decomposition groups

Published on 13 Apr 2026 - 18:14

91

Views

Simple proof for independence of square roots of two primes

Published on 14 Apr 2026 - 4:15

92

Views

Must a field extension contain an element of particular degree?

Published on 13 Apr 2026 - 22:53

81

Views

End = Aut $\Rightarrow$ algebraic?

Published on 14 Apr 2026 - 7:37

266

Views

Using totally ramified prime to find the degree of extension of cyclotomic field.

Published on 25 Mar 2026 - 6:30

79

Views

Signature function in field extension is onto

Published on 15 Apr 2026 - 17:47

489

Views

Dimension of commutator of an $n \times n$ matrix $A$ is $n$ implies that min poly and ch poly of $A$ coincides.

Published on 15 Apr 2026 - 8:02

77

Views

Proof that there exists an unramified extension of a particular defree.

Published on 13 Apr 2026 - 0:26

601

Views

An example of a field extension which is not Galois.

Published on 11 Apr 2026 - 8:37

463

Views

Adjoining Elements: The field $F(S)$

Published on 15 Apr 2026 - 0:00

List Question

Is there a polynomial which detects when the two smallest roots of a given real polynomial are equal?

How to show that if $\pi(x) \mid x^{p^n} - x$ then $\deg(\pi(x)) \mid n$?

Finding $f= g(s_1,s_2,s_3)$ if $char(K) = p$

Cartesian product of a field as a ring extension

Finding splitting fields over $\mathbb{Q}(\sqrt{-3})$

Intersection of Decomposition groups

Simple proof for independence of square roots of two primes

Must a field extension contain an element of particular degree?

End = Aut $\Rightarrow$ algebraic?

Using totally ramified prime to find the degree of extension of cyclotomic field.

Signature function in field extension is onto

Dimension of commutator of an $n \times n$ matrix $A$ is $n$ implies that min poly and ch poly of $A$ coincides.

Proof that there exists an unramified extension of a particular defree.

An example of a field extension which is not Galois.

Adjoining Elements: The field $F(S)$

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