Cyclotomy and arithmetic properties of some families in Z[x]/〈xpq − 1〉

Abstract

Cyclotomic classes of order 2 with respect to a product of two distinct odd primes [Formula: see text] and [Formula: see text] are represented in some specific forms and using these forms an alternate proof of Theorem 3 of [C. Ding and T. Helleseth, New generalized cyclotomy and its applications, Finite Fields Appl. 4 (1998) 140–166] is given, when [Formula: see text]. Further, it is observed that these classes are related to [Formula: see text]-cyclotomic cosets, where [Formula: see text] and [Formula: see text] such that gcd([Formula: see text]. Finally, arithmetic properties of some families in [Formula: see text] and hence in [Formula: see text] are studied.