Definition. Let , and be integers of an algebraic number field and . One defines
if and only if , i.e. iff there is an integer of with
Theorem. The congruence “” modulo defined above is an equivalence relation in the maximal order of . There are only a finite amount of the equivalence classes, the residue classes modulo .
Proof. For justifying the transitivity of “”, suppose (1) and ; then there are the integers and of such that ,
. Adding these equations we see that with the integer of . Accordingly, .
Let be an arbitrary integer of and a minimal basis of the field. Then we can write
where the ’s are rational integers. For , the division algorithm determines the rational integers and with
So we have
where and are some integers of the field. If are the algebraic conjugates of , then
Hence, divides in the ring of integers of , and (2) implies
Since any number has different possible values , there exist different ordered tuplets . Therefore there exist at most different residues and residue classes in the ring.