A note on order one cyclotomic polynomials

Abstract

It is a well known fact that, if p is an odd prime, then the pth-elementary cyclotomic polynomial Φp(x) has an associated p-Eisenstein polynomial  Φp(x). We extend this construction and show that, every order one elementary cyclotomic polynomial  Φ₂8_pt(x) has an associated p-Eisenstein polynomial Φ₂8_pt(x). In addition, for each Φ₂8_pt(x) , we investigate the divisibility (with respect to the prime p) of the coefficients of Φ₂8_pt(x). We also establish analogous results for order one Carlitz cyclotomic polynomials over F_q[T].

Keywords: Eisenstein and Carlitz cyclotomic polynomials, logarithmic and Mahler
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