Let $R$ be a PID and $M$ a free, finite rank $R$-module with a perfect billinear form $\omega$ such that $\omega(v,v)=0$ for all $v \in M$. Does anyone know a reference for the fact that a symplectic basis exists? By symplectic basis, I mean a basis $x_1,\ldots x_n,y_1 \ldots y_n$ of $M$ with $\omega(x_i,x_j)=0$, $\omega(y_i,y_j)=0$, and $\omega(x_i,y_j)=\delta_{i,j}$.

## 1 Answer

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See, for example, Max-Albert Knus, Quadratic and Hermitian forms over rings, Chapter I, Corollary 4.1.2. Or just mimic a proof in the case of fields (note, however, that "perfect" here should mean that the determinant of the Gram matrix is invertible, not just non zero).