Question

Prove that the polynomial $P(x) = x^{p−1} + 2x^{p−2} + 3x^{p−3} +···+ (p − 1)x + p$ is irreducible in $\mathbb Z[x].$

score 79 · Answer 1 · 2021-06-18 14:02:22

79

Views

Prove that the polynomial $P(x) = x^{p−1} + 2x^{p−2} + 3x^{p−3} +···+ (p − 1)x + p$ is irreducible in $\mathbb Z[x].$

Published on 18 Jun 2021 - 14:02

score 129 · Answer 2 · 2021-06-18 15:34:16

129

Views

Proving a polynomial in $\mathbb{F}_p[x]$ is irreducible if and only if it is relatively prime to $x^{p^2}-x$.

Published on 18 Jun 2021 - 15:34

score 41 · Answer 3 · 2026-04-12 16:37:07

41

Views

$ p(x) $ as a product of irreducible (over $ \mathbb{F}$) factors.

Published on 12 Apr 2026 - 16:37

score 72 · Answer 4 · 2026-04-12 17:47:46

72

Views

Irreducible polynomial proof

Published on 12 Apr 2026 - 17:47

score 62 · Answer 5 · 2021-06-25 23:58:55

62

Views

Prove that any polynomial over a field in $n$ variables can be factored into irreducibles

Published on 25 Jun 2021 - 23:58

score 33 · Answer 6 · 2026-04-10 16:05:31

33

Views

Properties of Integer Polynomials taking prime values infinitely often.

Published on 10 Apr 2026 - 16:05

score 164 · Answer 7 · 2021-06-29 13:22:17

164

Views

Find the irreducible decomposition of $x^{16}+x$ over $\mathbb{F_8}=\mathbb{F_2}[t]/(t^{3}+t+1)$.

Published on 29 Jun 2021 - 13:22

score 509 · Answer 8 · 2021-06-30 04:10:58

509

Views

$x^4+6x^2+1$ is reducible modulo $p$ for every prime $p$.

Published on 30 Jun 2021 - 4:10

score 123 · Answer 9 · 2026-03-30 15:30:51

123

Views

Irreducibility of polynomials that almost satisfy the Eisenstein's Criterion for UFDs

Published on 30 Mar 2026 - 15:30

score 232 · Answer 10 · 2021-06-30 14:02:50

232

Views

M. Ram Murty's proof of Cohn's Irreducibility criteria.

Published on 30 Jun 2021 - 14:02

score 79 · Answer 11 · 2026-04-12 08:20:26

79

Views

degree of the extension of a function field

Published on 12 Apr 2026 - 8:20

score 143 · Answer 12 · 2026-04-12 14:21:32

143

Views

Irreducibility of a two variables polynomial

Published on 12 Apr 2026 - 14:21

score 64 · Answer 13 · 2026-04-10 18:39:57

64

Views

Cohn's Irreducibility criterion in function fields over finite field

Published on 10 Apr 2026 - 18:39

score 95 · Answer 14 · 2021-07-03 15:37:34

95

Views

Proving that $(x^2 + 2)^n + 5(x^{2n-1} + 10x^2 + 5)$ is irreducible

Published on 03 Jul 2021 - 15:37

score 326 · Answer 15 · 2026-04-13 03:58:09

326

Views

Sufficient condition for irreducibility of homogeneous polynomial (Exercise I. 5.9 in Hartshorne)

Published on 13 Apr 2026 - 3:58

List Question

Prove that the polynomial $P(x) = x^{p−1} + 2x^{p−2} + 3x^{p−3} +···+ (p − 1)x + p$ is irreducible in $\mathbb Z[x].$

Proving a polynomial in $\mathbb{F}_p[x]$ is irreducible if and only if it is relatively prime to $x^{p^2}-x$.

$ p(x) $ as a product of irreducible (over $ \mathbb{F}$) factors.

Irreducible polynomial proof

Prove that any polynomial over a field in $n$ variables can be factored into irreducibles

Properties of Integer Polynomials taking prime values infinitely often.

Find the irreducible decomposition of $x^{16}+x$ over $\mathbb{F_8}=\mathbb{F_2}[t]/(t^{3}+t+1)$.

$x^4+6x^2+1$ is reducible modulo $p$ for every prime $p$.

Irreducibility of polynomials that almost satisfy the Eisenstein's Criterion for UFDs

M. Ram Murty's proof of Cohn's Irreducibility criteria.

degree of the extension of a function field

Irreducibility of a two variables polynomial

Cohn's Irreducibility criterion in function fields over finite field

Proving that $(x^2 + 2)^n + 5(x^{2n-1} + 10x^2 + 5)$ is irreducible

Sufficient condition for irreducibility of homogeneous polynomial (Exercise I. 5.9 in Hartshorne)

Trending Questions

Popular # Hahtags

Popular Questions