K2 of p-Adic Group Rings of Abelian p-Groups.
Let be a field of degree over , the field of rational -adic numbers, say with residue degree , ramification index and differential exponent . Let be the ring of integers of and its unique prime ideal. The trace and norm maps for are denoted and , respectively. Fix , a power of a prime , and let be a numerical character defined modulo and of order . The character extends to the ring of -adic integers in the natural way; namely , where denotes the residue class...
We illustrate how a particular expression, involving the generalized Bernoulli polynomials, satisfies systems of congruence relations if and only if a similar expression, involving the generalized Bernoulli numbers, satisfies the same congruence relations. These congruence relations include the Kummer congruences, and recent extensions of the Kummer congruences provided by Gunaratne.