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. ).

Now a solution in the form to is a unit in the ring . For , denote its conjugate by . Then we have the multiplicative norm

.

It follows that the solutions to form a subgroup of the multiplicative group of units of .

**Case 1:** . Then defines an ellipse, so there are at most finitely many solutions, i.e. is a finite subgroup of . Hence is cyclic and every solution is of the form

.

**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 .

**Corollary. **(Dirichlet’s unit theorem for a real quadratic field) If is a real quadratic field, then for some .

### Like this:

Like Loading...

*Related*