Let A $\in$ K$^n$$^,$$^n$ be a symmetric matrix over a field K. Define f(A) = { t$^2$ det(A) : t $\in$ K } and g(A) = {$x^T$A$x$ : $x \in $ K$^n$ } .
I have shown that if A,B are congruent, then f(A) = f(B) and g(A) = g(B).
Suppose A is the matrix of a quadratic form q with respect to some basis. Then we can write f(q) = f(A) and g(q) = g(A). Put K = $Z_3$, the field of three elements. Now using the foregoing, I want to find the congruence classes of the quadratic forms in two variables $x, y$ over K and their f- and g-values. (Two quadratic forms are congruent if they differ by a change of coordinates)
Can someone help?