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irreducible in Q[x]

Source: 2023 Viet Nam math olympiad for high school students D2 P6

March 25, 2023
algebrapolynomialnumber theoryprime numbers

Problem Statement

Prove that these polynomials are irreducible in Q[x]:\mathbb{Q}[x]: a) xpp!+xp−1(p−1)!+...+x22+x+1,\frac{{{x^p}}}{{p!}} + \frac{{{x^{p - 1}}}}{{(p - 1)!}} + ... + \frac{{{x^2}}}{2} + x + 1, with pp is a prime number. b) x2n+1,x^{2^n}+1, with nn is a positive integer.
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