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

Asked Jun 18 '21 at 14:02

Active Jun 18 '21 at 14:02

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

The Eisenstein-Schönemann theorem cannot be used to conclude anything at this stage. Is there some other method to prove this? This was a question that required only elementary mathematics i.e. fields etc. are not required.

asked Jun 18 '21 at 14:02

mathx

For primes $p = 1 \mod 4$ we can use $\mod 2$ test. The other case remains to be solved. – Infinity_hunter Jun 18 '21 at 14:14
2

Or this. – Jyrki Lahtonen Jun 18 '21 at 14:23
@JyrkiLahtonen Easy! Thanks. Could you elaborate about what the first comment to this question is trying to refer to? – mathx Jun 18 '21 at 14:33
@Infinity_hunter Could you elaborate a little more? – mathx Jun 18 '21 at 14:34
I'm afraid I don't have a clue what the first comment means. When reduced modulo $2$ the polynomial becomes a perfect square in $\Bbb{Z}_2[x]$ modulo every prime, so cannot be that. – Jyrki Lahtonen Jun 18 '21 at 14:36

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

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