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Irreducible rational polynomial

Source: Romania TST 5 2010, Problem 3

August 25, 2012
algebrapolynomialIrreducibleirreducibility

Problem Statement

Let pp be a prime number,let n1,n2,…,npn_1, n_2, \ldots, n_p be positive integer numbers, and let dd be the greatest common divisor of the numbers n1,n2,…,npn_1, n_2, \ldots, n_p. Prove that the polynomial Xn1+Xn2+⋯+Xnp−pXd−1\dfrac{X^{n_1} + X^{n_2} + \cdots + X^{n_p} - p}{X^d - 1} is irreducible in Q[X]\mathbb{Q}[X].
Beniamin Bogosel
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