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f(x^n) is irreducible

Source: Iran 3rd round 2011-algebra exam-p5

September 7, 2011
algebrapolynomialtrigonometryalgebra proposed

Problem Statement

f(x)f(x) is a monic polynomial of degree 22 with integer coefficients such that f(x)f(x) doesn't have any real roots and also f(0)f(0) is a square-free integer (and is not 11 or āˆ’1-1). Prove that for every integer nn the polynomial f(xn)f(x^n) is irreducible over Z[x]\mathbb Z[x].
proposed by Mohammadmahdi Yazdi
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