The Artin-Hasse power series and p-adic group rings

Abstract

If p is an odd prime and A an abelian p-group, consider the multiplicative group formed by {1 + ∑ z ∈ A a z(z − 2 + z −1) | a z ∈ Z p } in the p-adic group ring. This group is shown to be a Z p -module with the basis { E( z − 2 + z −1) | 1 ≠ z ∈ A}, where E( T) denotes the well-known p-adically integral power series exp(Σ n ≥ 0 p − n T p n ).