On the degrees of irreducible factors of a polynomial

Abstract

Assume that f ( x ) = a n x n + a n − 1 x n − 1 + ⋯ + a 0 , a 0 ≠ 0 f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0, a_0\neq 0 is a polynomial with rational coefficients and let p p be a prime number whose highest power r i r_i dividing a i a_i (where r i = ∞ r_i = \infty if a i = 0 a_i = 0 ) satisfies r n = 0 r_n = 0 , r i ≥ 1 r_i \geq 1 for 0 ≤ i ≤ n − 1 0\leq i\leq n-1 . In this article, we explicitly construct a number d d depending only on r i r_i ’s and show that each irreducible factor of f ( x ) f(x) has degree at least d d over Q \mathbf {Q} . This result extends the famous Eisenstein-Dumas irreducibility criterion. In fact, we prove our result in a more general setup for polynomials with coefficients in an arbitrary valued field.