Let be the ring of integers in a fixed finite extension , and let be a nonzero ideal in . Prove that contains infinitely many rational integers.
We begin with a lemma.
Lemma: Let be an ideal. If , then , where denotes the field norm. Proof: Note that since is a rational integer. So . Moreover, by definition, is an algebraic integer, being the product of the (certainly integral) conjugates of for . So . Since is an ideal, .
Now if , then there exists a nonzero element . We have nonzero, and thus for all . In particular, contains all of the -multiples of .