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

### Volume 167 / 2015

Acta Arithmetica 167 (2015), 281-298
MSC: Primary 11R29.
DOI: 10.4064/aa167-3-6

#### Abstract

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 ${\mathbb {Q}_p}$ of the field of rational numbers $\mathbb {Q}$. Additional connections to the Gross–Koblitz formula and explicit congruences for quadratic and cubic fields are given.