Let be an algebraic number field with ring of integers , and let be and ideal. Suppose is the smallest positive integer contained in . Show that . Suppose now that is a rational prime; show that .
By Theorem 9.16, . Now is an ideal of , which is principal since is a principal ideal domain. Certainly then is the smallest positive integer in . Thus .
Now suppose is a rational prime. Suppose there is a (minimal) integer such that . By part (1), , a contradiction. So none of the integers in are in , and thus mod for all . Moreover, we have mod for all with , since otherwise (assuming ) we have mod . Since , the cosets exhaust and are mutually exclusive.