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P(x^2) is irreducible in Q[x]

Source: 2023 Viet Nam math olympiad for high school students D2 P7

March 25, 2023
algebra

Problem Statement

Given a polynomial with integer coefficentsP(x)=xn+an−1xn−1+...+a1x+a0,n≥1P(x)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0,n\ge 1satisfying these conditions: i) ∣a0∣|a_0| is not a perfect square. ii) P(x)P(x) is irreducible in Q[x].\mathbb{Q}[x].
Prove that: P(x2)P(x^2) is irreducible in Q[x].\mathbb{Q}[x].
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