We want to find the solutions to
over the integers, where .
Let and be the roots of . We can factorise as . If is rational, it must be an integer and then it is easy to find all solutions. So let us consider the more interesting case where is not rational (i.e. ).
Case 2: . Now is real, so is a subgroup of .
Claim. is not dense.
Proof. WLOG . Then for we have
Suppose that is a solution. Then
impossible. Thus , i.e. is not dense.
It follows that is either or for some (cf. ordering in groups, corollary 4). Therefore all solutions to are either just , or are given by
where is the least solution .