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Let m ≥ 1 and a m be integers. Let α be a rational number which is not a negative integer such that α = u v with gcd(u, v) = 1, v > 0. Let φ(x) belonging to Z[x] be a monic polynomial which is irreducible modulo all the primes less than or equal to vm + u. Let a i (x) with 0 ≤ i ≤ m − 1 belonging to Z[x] be polynomials having degree less than deg φ(x). Assume that the content of (a m a 0 (x)) is not divisible by any prime less than or equal to vm + u. In this paper, we prove that the polynomialsb j a j (x)φ(x) j ) are irreducible over the rationals for all but finitely many m, where bis irreducible over rationals for each α ∈ {0, 1, 2, 3, 4} unless (m, α) ∈ {(1, 0), (2, 2), (4, 4), (6, 4)}. For proving our results, we use the notion of φ-Newton polygon and some results from analytic number theory. We illustrate our results through examples.

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].