A note on order one cyclotomic polynomials

Abstract

Abstract. It is a well known fact that, if p is an odd prime, then the pth-elementary cyclotomic polynomial _p(x) has an associated p-Eisenstein polynomial c_p(x). We extend this construction and show that, every order one elementary cyclotomic poly-nomial _2spt (x) has an associated p-Eisenstein polynomial\_2spt (x). In addition, for each _2spt (x), we investigate the divisibility (with respect to the prime p) of the coefficients of\_2spt (x). We also establish analogous results for order one Carlitz cyclotomic polynomials over Fq[T].