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On an Irreducibility Theorem of A. Cohn

1981; Cambridge University Press; Volume: 33; Issue: 5 Linguagem: Inglês

10.4153/cjm-1981-080-0

1496-4279

John Brillhart, Michael Filaseta, Andrew Odlyzko,

Meromorphic and Entire Functions

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 as then the polynomial irreducible in Z [ 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 α 1 , α 2 , …, α n . 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 in Z [ x ].