The discriminant is an invariant of a general symmetric bilinear form on a vector space $V$ over a field $K$. I'm trying to understand for which fields the discriminant is a complete invariant.
First i'll consider the two dimensional case. Say that we have a diagonalized form have nonzero diagonal entries $a,b$. Then its discriminant is $ab$. A second diagonalized form with the same discriminant has entries $1$ and $ab$. We need that there is a change of basis between these two diagonalizations.
Say that the basis of the first form is $u_1,u_2$. Then $(u_1,u_1)=a$, $(u_2,u_2)=b$ and $(u_1,u_2)=0$. If $v_1$ is the first basis element of the second basis, we have that $(v_1,v_1)=1$. Say $v=xu_1+yu_2$. Then $1=(v_1,v_1)=ax^2+by^2$.
Assuming such a solution exists, how would we define the second basis element $v_2$ of the second basis so that $(v_2,v_2)=ab$?
After I figure out how to define $v_2$, I believe that the discriminant will be a complete invariant of the form as long as $\forall a,b \in K$ the equation $1=ax^2+by^2$ has a solution.
Is there a way I could generalize this to dimensions higher than $2$?
Thanks! Also, I wouldn't mind general insights into what's going on here in addition to my specific questions. Thanks again!
Is this reasoning all clear? How would I generalize this to dimensions higher than $2$? Thanks