Decomposition of congruences involving a map Φ

Congruences of Ankeny-Artin-Chowla type modulo p² for a cyclic subfield K of prime conductor p were derived by Jakubec and expressed in terms of a technically defined map Φ. Later, Jakubec and Lassak found a decomposition of the map Φ modulo p² and simplified the formulation of these congruences. A corresponding decomposition of the map Φ modulo p³ was obtained in [MARKO, F.: Towards Ankeny-Artin-Chowla type congruence modulo p³, Ann. Math. Sil. 20 (2006), 31-55]. That technical step was important for the formulation of congruences of Ankeny-Artin-Chowla type modulo p³. This paper will show how to produce an analogous decomposition of the map Φ modulo an arbitrary power pⁿ which would allow a description of analogous congruences modulo pⁿ.