Thank you. This is the kind of thing I was looking for. But is there a corresponding extension of this method to the case of translations or "affine modules"?

Some useful references on trace forms are: Conner, Perlis - A survey of trace forms of algebraic number fields, and Mantilla-Soler - On the arithmetic determination of the trace.

The fact that the inclusion $\mathbb{Z} \subset \mathbb{Q}$ does not preserve dimension can be expressed by saying that $\mathbb{Q}$ is zero-dimensional, but not hereditarily zero dimensional. These are studied in the book Zero-Dimensional Commutative Rings, edited by David Dobbs.

@wishcow Pete Clark has a nice exposition of this theorem and other aspects of quadratic forms. Check p22 of math.uga.edu/%7Epete/quadraticforms.pdf