My goal for the problem is to classify all equivalence classes of anisotropic quadratic forms over \mathbb{F}_3((t)) and then use that to determine 1) which of these forms stay anistropic and 2) which become isometric to each other over:

1. \mathbb{F}_9((t))
2. \mathbb{F}_3((\sqrt{t}))

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 u(\mathbb{F}_3((t))=4) are:

\langle1\rangle, \langle2\rangle, \langle t\rangle, \langle2t\rangle
\langle1,1\rangle, \langle t,t\rangle, \langle1,t\rangle, \langle2,t\rangle
\langle1,1,t\rangle,\langle1,1,2t\rangle,\langle1,  t,t\rangle,\langle2,t,t\rangle

where the notation \langle a,b\rangle, for example, means the quadratic form q(x)=ax^2+by^2.

1. Over \mathbb{F}_9((t)), 2 becomes a square, so we have

\langle1\rangle\cong\langle2\rangle, \langle t\rangle\cong\langle2t\rangle
\langle1,1\rangle, \langle t,t\rangle, \langle1,t\rangle\cong\langle2,t\rangle
\langle1,1,t\rangle\cong\langle1,1,2t\rangle, \langle 1,t,t\rangle\cong\langle2,t,t\rangle

Nothing becomes isotropic.

2. Over \mathbb{F}_3((\sqrt{t})), t becomes a square, so all that remains is

\langle1\rangle\cong\langle t\rangle,\langle2\rangle\cong\langle2t\rangle
\langle1,1\rangle\cong\langle t,t\rangle\cong\langle1,t\rangle

Everything else becomes isotropic.