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Irreducible polynomial

Source: Chinese TST

April 9, 2008
algebrapolynomialmodular arithmeticabsolute valuealgebra proposed

Problem Statement

Prove that for all n≥2, n\geq 2, there exists n n-degree polynomial f(x) \equal{} x^n \plus{} a_{1}x^{n \minus{} 1} \plus{} \cdots \plus{} a_{n} such that (1) a1,a2,⋯ ,an a_{1},a_{2},\cdots, a_{n} all are unequal to 0 0; (2) f(x) f(x) can't be factorized into the product of two polynomials having integer coefficients and positive degrees; (3) for any integers x,∣f(x)∣ x, |f(x)| isn't prime numbers.
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