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 Φ₂^s_p(x) has an associated p-Eisenstein polynomial . We extend this construction and show that, every order one elementary cyclotomic polynomial Φ₂^s p^t (x) has an associated p-Eisenstein polynomial . In addition, for each Φ₂^s p^t (x), we investigate the divisibility (with respect to the prime p) of the coefficients of . We also establish analogous results for order one Carlitz cyclotomic polynomials over _q [T ].

Key words:

• Eisenstein and Carlitz cyclotomic polynomials
• logarithmic and Mahler heights

Mathematics Subject Classification (2010):

• 11A99
• 11G09