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Eisenstein's criterion

Source: 2023 Viet Nam math olympiad for high school students D2 P3

March 25, 2023
algebranumber theoryprime numbers

Problem Statement

Given a polynomial with integer coefficents with degree n>0:n>0:P(x)=anxn+...+a1x+a0.P(x)=a_nx^n+...+a_1x+a_0. Assume that there exists a prime number pp satisfying these conditions: i) p∣aip|a_i for all 0≤i<n,0\le i<n, ii) p∤an,p\nmid a_n, iii) p2∤a0.p^2\nmid a_0. Prove that P(x)P(x) is irreducible in Z[x].\mathbb{Z}[x].
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