Number of equivalence classes regarding congruency of symmetric bilinear forms Let $n \in \mathbb{N}$. Let $V$ be a $n$-dimensional vector space over a field $K$. Consider the set of symmetrical bilinear forms over $V$ with the equivalence relation of congruency. Two bilinear forms are congruent $\iff$ there is an ordered basis $\mathfrak{B}$ of $V$ such that the representing matrices of the bilinear forms are congruent. The task is to find out the number of equivalence classes for fields $\mathbb{R}, \mathbb{C}, \mathbb{Q}$. However I struggle right at the beginning. I have found out that the equivalence relation also means that there exist two ordered bases $\mathfrak{B_1}, \mathfrak{B_2}$ such that the representative matrices of the bilinear forms are equal. Does that help me? I am stuck.

Translating this to a less abstract language.

If $n$ is finite the set of symmetric forms corresponds to the set of $n\times n$ symmetric matrices.

Two symmetric forms are equivalent if there is a basis that makes the two congruent.

There exists a $P$ such that $A = P^{-1} B P$

And since all symmetric matrices are diagonalizable.

i.e. $Q^{-1} A Q = P^{-1} B P = D$

Two matrices are equivalent if they have the same eigenvalues.