Characterize the units in .
Note that every element of is real; in particular, every element is either positive or negative. Note also that is a unit with inverse .
We begin with a lemma.
Lemma: No unit in satisfies . Proof: Suppose to the contrary that is such a unit. Now , so that . Since , we have . Adding these inequalities, we have , and thus . So . Since , we have . Consider again the inequality ; for in this range, we have no integer solutions .
Now let be a unit; say . Since , there exists such that . Now , and by the lemma (since is a unit), .
Thus every unit in is an integer power of .