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Perron's criterion

Source: 2023 Viet Nam math olympiad for high school students D2 P5

March 25, 2023
algebra

Problem Statement

Given a polynomialP(x)=xn+an−1xn−1+...+a1x+a0∈Z[x]P(x)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0\in \mathbb{Z}[x] with degree n≥2n\ge 2 and ao≠0.a_o\ne 0. Prove that if ∣an−1∣>1+∣an−2∣+...+∣a1∣+∣a0∣|a_{n-1}|>1+|a_{n-2}|+...+|a_1|+|a_0|, then P(x)P(x) is irreducible in Z[x].\mathbb{Z}[x].
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