Let be a quadratic number field. For , recall that the discriminant is defined as
where are the Galois conjugates of and are those of . For any we define its discriminant to be .
Write and . Then
If are the Galois conjugates of , then
Now suppose that . Then is spanned by , so there are integers such that
So we have
Lemma. If is a matrix with integer coefficients and with , then if and only if , the identity matrix.
Proof. This follows from the -linear independence of . More concretely,
Thus , so that . But and are integers. Hence , i.e. . Thus
Fact. spans if and only if .
Note that all of the above arguments generalize to arbitrary number fields.
A nice corollary:
Corollary. and generate (as a group) if and only if