Units in some group rings over the ring of p-cyclotomic integers

Abstract

Describing the group of units of a group ring is a classical problem. Let [Formula: see text] be a rational prime number. We set [Formula: see text] a primitive root of unity of order [Formula: see text], [Formula: see text] the ring of [Formula: see text]-cyclotomic integers, [Formula: see text] a finite abelian [Formula: see text]-group and [Formula: see text] the group of the units [Formula: see text] of [Formula: see text] such that [Formula: see text], where [Formula: see text] is the augmentation map. We will prove that all the elements of the group [Formula: see text] arise from the units of the group ring [Formula: see text], where [Formula: see text] is the cyclic group of order [Formula: see text]. As an application, we describe explicitly the group of units of the group ring [Formula: see text] when [Formula: see text] is an elementary abelian [Formula: see text]-group and [Formula: see text] is a regular prime number.