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

.

In other words, two bases generate the same lattice only if their fundamental parallelograms have equal areas.

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