On the factorization of polynomials with small Euclideannorm

Throughout this paper, we refer to the non-cyclotomic part of a polynomial f(x) 2 Z[x] as f(x) with its cyclotomic factors removed. More speci cally, if g1(x); : : : ; gr(x) are non-cyclotomic irreducible polynomials in Z[x] and gr+1(x); : : : ; gs(x) are cyclotomic polynomials such that f(x) = g1(x) gr(x) gr+1(x) gs(x), then g1(x) gr(x) is the non-cyclotomic part of f(x). We refer to a polynomial f(x) 2Z[x] of degree n as reciprocal if f(x) = xf(1=x). We refer to xf(1=x) as the reciprocal of f(x). Analogous to our rst de nition, we refer to the non-reciprocal part of f(x) 2Z[x] as f(x) with the irreducible reciprocal factors having positive leading coe cient removed. Here and throughout this paper we refer to irreducibility over the integers so that the irreducible polynomials under consideration have integer coe cients and content one. Observe that a reciprocal polynomial may be equal to its non-reciprocal part as is the case, for example, with x + x + x + 3x + x + x+ 1 which factors as a product of two non-reciprocal irreducible polynomials. In 1956, E.S. Selmer [8] investigated the irreducibility over the rationals of the trinomials x + "1x a + "2 where n > a > 0 and each "j 2 f 1; 1g. He obtained complete solutions in the case a = 1 and partial results for a > 1. In 1960, W. Ljunggren [2] extended Selmer's work to deal generally with the case when a 1. In addition, he studied the quadrinomials x+ "1x+ "2x+ "3 where each "j 2 f 1; 1g and n > b > a > 0. There was a correctable error in Ljunggren's work involving the omission of certain cases; this was noted in 1985 by W.H. Mills [3] who lled in the gaps of Ljunggren's arguments. It was established that the noncyclotomic parts of the trinomials above are irreducible or, in the case that every factor is cyclotomic, identically 1. (Throughout this paper we view the polynomials 1 as neither reducible nor irreducible.) In the case of quadrinomials, the analo-