On an Irreducibility Theorem of A. Cohn

Abstract

In [1, b.2, VIII, 128] Pólya and Szegö give the following interesting result of A. Cohn:THEOREM 1. If a prime p is expressed in the decimal system asthen the polynomial irreducible inZ[x].The proof of this result rests on the following theorem of Pólya and Szegö [1, b.2, VIII, 127] which essentially states that a polynomial f(x) is irreducible if it takes on a prime value at an integer which is sufficiently far from the zeros of f(x).THEOREM 2. Let f(x) ∊ Z[x] be a polynomial with the zeros α1, α2, …, αn. If there is an integer b for which f(b) is a prime, f(b – 1) ≠ 0, and for 1 ≦ i ≦ n, then f(x) is irreducible inZ[x].