On the cyclotomic Dedekind embedding and the cyclic Wedderburn embedding

Let n >= 1 and let p be a prime. Let t = 1 - zeta_{p^n}. Expand an integer j in [0,p^n-1], coprime to p, p-adically as j = sum_{s >= 0} a_s p^s. Denote the tensor product over Z_(p) by o . Then the #([0,j] - (p))th Z_(p)[t]-linear elementary divisor of the cyclotomic Dedekind embedding Z_(p)[t] o Z_(p)[t] --> prod_{i in (Z/p^n)^*} Z_(p)[t] has valuation -1 + sum_{s >= 0} (a_s (s+1) - a_{s+1} (s+2)) p^s at t. There is a similar result for the related cyclic Wedderburn embedding.