| I. Schur established four theorems concerning the irreducibility (over the rationals) of the polynomials, f (x) = j=0naj xjj!,g x=j=0n ajx2ju2j ,hx= j=0najxj j+1!, and wx=j=0 najx2ju 2j+2 (a0, a1,..., an denote arbitrary integers satisfying |a n| = |a0| = 1 and for j ≥ 0, u2j = 1 x 3 x 5 x &cdots; x (2j - 1)) . For example, Schur proved that f(x) is irreducible for all positive integers n. Michael Filaseta generalized this result of Schur's by investigating the irreducibility of f(x) when the condition |an| = 1 is relaxed.;In this work, we generalize Schur's other three irreducibility theorems by relaxing the condition |an| = 1 to 0 < | an| < cn where cn is a constant depending on n. We use Newton polygons and analytic results concerning the distribution of primes in investigating the irreducibility of the above polynomials. We establish the following.;Generalization One. For 0 < |an| < 2n - 1, g(x) is irreducible for all but finitely many pairs (an, n). Furthermore, for every integer n > 1, if |a n| = 2n - 1 and |a 0| = 1, then there exist integers an -1, an-2,..., a1 such that g(x) is reducible.;Generalization Two. Let k' be the integer such that n + 1 = k'2 u where u is an integer ≥ 0 and ( k', 2) = 1. Let k″ be the integer such that (n + 1)n = k″2u3 v where (k″, 6) = 1, u is an integer ≥ 1, and v is an integer ≥ 0. Let M = min{k', k″}. Then for 0 < |an| 2, if |a n| = M and |a0| = 1, then there exist integers an-1 , an-2,..., a1 such that h(x) is reducible.;Generalization Three. Let k' be the integer such that 2n + 1 = k '3u where u is an integer ≥ 0 and (k', 3) = 1. Let k″ be the integer such that (2 n + 1) (2n - 1) = k ″3u5v where u and v are integers ≥ 0 and ( k″, 15) = 1. Let M = min{ k', k″}. Then for 0 < |an| 2, if |an| = M and |a0| = 1, then there exist integers a n-1, an-2 ,...,a1 such that w( x) is reducible. |
