Congruences of Ankeny-Artin-Chowla type and the p-adic class number formula revisited

The purpose of this paper is to interpret the results of Jakubec and his collaborators on congruences of Ankeny-Artin-Chowla type for cyclic totally real fields as an elementary algebraic version of the,p-adic class number formula modulo powers of p. We show how to generalize the previous results to congruences modulo arbitrary powers p^t and to equalities in the p-adic completion double-struck Q_p of the field of rational numbers double-struck Q. Additional connections to the Gross-Koblitz formula and explicit congruences for quadratic and cubic fields are given.