The structure of the n-th roots of unity in residue rings of prime ideals P over p in algebraic number fields

摘要

This paper is the first of a series of papers in which we shall determinethe solutions over the ring of integers OK of the congruencexn ≡ 1 (mod M), where M is an ideal in OK and n is a positive integer.This paper deals with the solutions of xn ≡ 1 (mod Pa), where a is apositive integer, P denotes a prime ideal in OK lying over the rationalprime number p and p - n.