Question

Show that $\mathbb{Q}(\sqrt[3]{2}, e^{2 \pi i/3}) = \mathbb{Q}(\sqrt[3]{2} + e^{2 \pi i/3})$

score 150 · Answer 1 · 2020-12-30 14:56:50

150

Views

Show that $\mathbb{Q}(\sqrt[3]{2}, e^{2 \pi i/3}) = \mathbb{Q}(\sqrt[3]{2} + e^{2 \pi i/3})$

Published on 30 Dec 2020 - 14:56

score 90 · Answer 2 · 2020-12-30 18:47:56

90

Views

Prove that polynomial $1+x+x^2$ of $\mathbb{R}[x]$ is irreducible over $\mathbb{R}$. Do not use the Eisenstein criterion.

Published on 30 Dec 2020 - 18:47

score 80 · Answer 3 · 2020-12-30 20:23:07

80

Views

Show that $x^4-4$ isn't reducible over $\Bbb Q$?

Published on 30 Dec 2020 - 20:23

score 42 · Answer 4 · 2021-01-01 16:23:00

42

Views

Consider the quadratic integer ring $A=\Bbb Z[\sqrt-2]$. Show that $U(A)=\{-1,1\}$.

Published on 01 Jan 2021 - 16:23

score 116 · Answer 5 · 2021-01-01 21:08:49

116

Views

$2$ is prime, but $2=(1+i)(1-i)$ is composite.

Published on 01 Jan 2021 - 21:08

score 82 · Answer 6 · 2021-01-04 21:32:27

82

Views

Let $f$ be an irreducible monic polynomial of degree 4 over $\mathbb{F}_p$. Is $f$ irreducible over $\mathbb{F}_{p^2}$?

Published on 04 Jan 2021 - 21:32

score 170 · Answer 7 · 2021-01-05 04:24:30

170

Views

All Polynomials Such That All Roots of $f(x)$ are in $f(x^2)$

Published on 05 Jan 2021 - 4:24

score 139 · Answer 8 · 2021-01-05 10:10:05

139

Views

How do we check that a polynomial is irreducible in $\mathbb{C}[x,y]$

Published on 05 Jan 2021 - 10:10

score 111 · Answer 9 · 2026-04-11 03:09:27

111

Views

Are the irreducible rational polynomials of given degree dense?

Published on 11 Apr 2026 - 3:09

score 82 · Answer 10 · 2021-01-06 17:17:23

82

Views

For odd $n$ is the polynomial $X^{n-1}-X^{n-2}+...-X+1$ irreducible in $\mathbb{Q}[X]$?

Published on 06 Jan 2021 - 17:17

score 42 · Answer 11 · 2026-04-10 02:44:05

42

Views

Galois Group of a Polynomial Problem help

Published on 10 Apr 2026 - 2:44

score 387 · Answer 12 · 2026-04-02 17:29:20

387

Views

Factorization of Polynomial in an extension field

Published on 02 Apr 2026 - 17:29

score 37 · Answer 13 · 2021-01-12 09:30:52

37

Views

Investigate if the following polynomial in Q [x] is reducible or irreducible: $x^3+6x^2+8x+4$

Published on 12 Jan 2021 - 9:30

score 60 · Answer 14 · 2021-01-13 04:48:31

60

Views

Question involving polynomials and fields

Published on 13 Jan 2021 - 4:48

score 34 · Answer 15 · 2021-01-13 10:22:14

34

Views

Let $p(x)$ be polynomial over $GF(2)$ with $deg(p(x))=8$, show that $g(x)\nmid x^{31}-1$

Published on 13 Jan 2021 - 10:22

List Question

Show that $\mathbb{Q}(\sqrt[3]{2}, e^{2 \pi i/3}) = \mathbb{Q}(\sqrt[3]{2} + e^{2 \pi i/3})$

Prove that polynomial $1+x+x^2$ of $\mathbb{R}[x]$ is irreducible over $\mathbb{R}$. Do not use the Eisenstein criterion.

Show that $x^4-4$ isn't reducible over $\Bbb Q$?

Consider the quadratic integer ring $A=\Bbb Z[\sqrt-2]$. Show that $U(A)=\{-1,1\}$.

$2$ is prime, but $2=(1+i)(1-i)$ is composite.

Let $f$ be an irreducible monic polynomial of degree 4 over $\mathbb{F}_p$. Is $f$ irreducible over $\mathbb{F}_{p^2}$?

All Polynomials Such That All Roots of $f(x)$ are in $f(x^2)$

How do we check that a polynomial is irreducible in $\mathbb{C}[x,y]$

Are the irreducible rational polynomials of given degree dense?

For odd $n$ is the polynomial $X^{n-1}-X^{n-2}+...-X+1$ irreducible in $\mathbb{Q}[X]$?

Galois Group of a Polynomial Problem help

Factorization of Polynomial in an extension field

Investigate if the following polynomial in Q [x] is reducible or irreducible: $x^3+6x^2+8x+4$

Question involving polynomials and fields

Let $p(x)$ be polynomial over $GF(2)$ with $deg(p(x))=8$, show that $g(x)\nmid x^{31}-1$

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