My goal for the problem is to classify all equivalence classes of anisotropic quadratic forms over and then use that to determine 1) which of these forms stay anistropic and 2) which become isometric to each other over:
I was hoping someone could look over this to see whether I'm going about this correctly.
The anisotropic equivalence classes for dimensions 1 through 4 (which is sufficient since ) are:
where the notation , for example, means the quadratic form .
1. Over , becomes a square, so we have
Nothing becomes isotropic.
2. Over , becomes a square, so all that remains is
Everything else becomes isotropic.