Question

Prove that $f(x)=2x^3+ax^2+bx+c$ is irreducible in $ℚ[x]$ if and only if $f({d\over2})≠0$ for all $a, b, c, d∈ℤ$.

score 248 · Answer 1 · 2026-04-16 22:02:15

248

Views

Prove that $f(x)=2x^3+ax^2+bx+c$ is irreducible in $ℚ[x]$ if and only if $f({d\over2})≠0$ for all $a, b, c, d∈ℤ$.

Published on 16 Apr 2026 - 22:02

score 32 · Answer 2 · 2026-04-13 07:07:03

32

Views

Express the fixed field of a subgroup $H$ of $\Gamma(L:K)$ in terms of a basis of $L$ over $K$.

Published on 13 Apr 2026 - 7:07

score 456 · Answer 3 · 2026-03-28 06:13:26

456

Views

Galois Group of $x^{6}-2x^{3}-1$

Published on 28 Mar 2026 - 6:13

score 105 · Answer 4 · 2026-04-16 18:58:57

105

Views

Roots of $x^{p^{n-1}}+\ldots+x^p+x$ in $\mathbb{F}_{p^n}$

Published on 16 Apr 2026 - 18:58

score 198 · Answer 5 · 2026-04-08 23:35:06

198

Views

Can Field of fractions be Algebraically Closed?

Published on 08 Apr 2026 - 23:35

score 89 · Answer 6 · 2026-04-10 03:08:29

89

Views

Is every field a field extension of some form.

Published on 10 Apr 2026 - 3:08

score 171 · Answer 7 · 2026-04-10 14:44:08

171

Views

Calculate order of multiplicative group of finite field

Published on 10 Apr 2026 - 14:44

score 1.1k · Answer 8 · 2026-04-12 22:49:18

1.1k

Views

Which field property enables us to multiply on both sides by the same value, while preserving equality?

Published on 12 Apr 2026 - 22:49

score 520 · Answer 9 · 2026-04-15 10:22:19

520

Views

Galois group of $x^p-a$ over $\mathbb{Q}$

Published on 15 Apr 2026 - 10:22

score 77 · Answer 10 · 2026-03-25 11:54:19

77

Views

Root of $f$ is $p^{\text{th}}$ power in extension field $\Rightarrow$ coefficients of $f$ are $p^{\text{th}}$ powers in base field.

Published on 25 Mar 2026 - 11:54

score 68 · Answer 11 · 2026-04-17 02:54:05

68

Views

Find basis of $\mathbb Q(\sqrt[3]{2},\sqrt{-3})$ over the field $\mathbb Q(\sqrt[3]{2}\omega)$.

Published on 17 Apr 2026 - 2:54

score 53 · Answer 12 · 2026-04-14 06:17:58

53

Views

$a$ is a norm in $K(\sqrt{a})/K$ if and only if it is the sum of two squares

Published on 14 Apr 2026 - 6:17

score 46 · Answer 13 · 2026-04-17 12:02:16

46

Views

Real or imaginary part of a field extension of $\mathbb{Q}$.

Published on 17 Apr 2026 - 12:02

score 1.2k · Answer 14 · 2026-03-25 03:01:37

1.2k

Views

Finding class number of quadratic number field using Minkowski bound

Published on 25 Mar 2026 - 3:01

score 30 · Answer 15 · 2026-04-14 17:46:21

30

Views

Question regarding p-basis in complete local ring

Published on 14 Apr 2026 - 17:46

List Question

Prove that $f(x)=2x^3+ax^2+bx+c$ is irreducible in $ℚ[x]$ if and only if $f({d\over2})≠0$ for all $a, b, c, d∈ℤ$.

Express the fixed field of a subgroup $H$ of $\Gamma(L:K)$ in terms of a basis of $L$ over $K$.

Galois Group of $x^{6}-2x^{3}-1$

Roots of $x^{p^{n-1}}+\ldots+x^p+x$ in $\mathbb{F}_{p^n}$

Can Field of fractions be Algebraically Closed?

Is every field a field extension of some form.

Calculate order of multiplicative group of finite field

Which field property enables us to multiply on both sides by the same value, while preserving equality?

Galois group of $x^p-a$ over $\mathbb{Q}$

Root of $f$ is $p^{\text{th}}$ power in extension field $\Rightarrow$ coefficients of $f$ are $p^{\text{th}}$ powers in base field.

Find basis of $\mathbb Q(\sqrt[3]{2},\sqrt{-3})$ over the field $\mathbb Q(\sqrt[3]{2}\omega)$.

$a$ is a norm in $K(\sqrt{a})/K$ if and only if it is the sum of two squares

Real or imaginary part of a field extension of $\mathbb{Q}$.

Finding class number of quadratic number field using Minkowski bound

Question regarding p-basis in complete local ring

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