Geodesic
On an Irreducibility Theorem of A. Cohn
Canadian journal of mathematics, Tome 33 (1981) no. 5, pp. 1055-1059

Voir la notice de l'article provenant de la source Cambridge University Press

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 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 α 1, α 2, ..., 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]. 
Brillhart, John; Filaseta, Michael; Odlyzko, Andrew. On an Irreducibility Theorem of A. Cohn. Canadian journal of mathematics, Tome 33 (1981) no. 5, pp. 1055-1059. doi: 10.4153/CJM-1981-080-0
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[1] 1. Pólya, G. and Szego, G., Aufgaben und Lehrsatze aus der Analysis, (Springer-Verlag, Berlin, 1964). Google Scholar

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