Canonical forms of matrices under congruence relation

Question

Let $A,B$ be square matrices over a field $K$ (We can assume $K$ to be finite, if needed). Consider the equivalence relation $A\sim B$ if and only if there exists an invertible matrix S such that $A=SBS^T$. What would be the canonical representative for an equivalence class? (i.e. I am essentially asking what is a normal form). Over $\mathbb R$, this is managed by the Sylvester law of inertia, but over other fields I suspect that a represent the equivalence class is given by a diagonal matrix with diagonal entries given by a certain number of non-squares, a certain number of squares different from zero, and a certain number of zeroes. Does anyone have a reference for this? Or maybe this is false?