On the factors of a polynomial

In this article, we show that if f ( x ) = a n x n + a n − 1x n − 1 + ⋯ + a 0 , a 0 ≠ 0 is a polynomial with rational coefficients and if there exists a prime p whose highest power r i dividing a i (wherer i = ∞ ifa i = 0 ) satisfiesr n = 0 , n r i ⩾ ( n − i ) r 0 > 0 for 0 ⩽ i ⩽ n − 1 , then f ( x ) has at most gcd ( r 0 , n ) irreducible factors over the field Q of rational numbers and each irreducible factor has degree at least n / gcd ( r 0 , n )over 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 arbitrary valued fields.