In this paper the results of [4] and [5] on the congruence of Ankeny-Artin-Chowla type modulo p2 for real subfields of the Q(ζp) of a prime degree l > 2 are simplified to explicit forms (2) and (3) of the Theorem 1. The congruence is then used to calculations of class numbers in special quintic fields and to some calculations in cubic fields.
We formulate and generalize the technique of Jakubec established to derive congruences of Ankeny-Artin-Chowla type for a cyclic subfleld K of prime conductor p. Then we concentrate on the case of congruences modulo p3 and clear a significant technical hurdle which allows us to formulate Ankeny-Artin-Chowla congruences modulo p3 in a concise way.
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