Let be a quadratic field, and let be an element with . Prove that there exists an algebraic integer such that , where denotes conjugate.
Let . If , then , and either or ; similarly if . So we assume that and . Now . Rearranging, we have . Choose some integers and such that is equal to this common ratio. Let ; certainly is an algebraic integer in . We claim that .
To that end, note that . From , we see that , and likewise since we have . So , and thus .