Let $V$ be a vector space over $k$, a field of characteristic $2$, I wonder how to show that in general there is no bijection between quadratic forms and symmetric bilinear forms.
I understand that since in this case $2$ is not invertible, the canonical correspondence fails, but why couldn't there be any bijection between those two sets in general?
Best Answer
In characteristic $2$ there is essentially the non-canonical (vector space) isomorphism from $n$-ary symmetric bilinear forms to $n$-ary quadratic forms $$\sum_{i,j} C_{ij} x_i y_j \mapsto \sum_{i \le j} C_{ij} x_i x_j \qquad (C_{ij} = C_{ji})$$ while the canonical map $\sum_{i,j} C_{ij} x_i y_j \mapsto \sum_{i, j} C_{ij} x_i x_j= \sum_i C_{ii} x_i^2$ is non-surjective nor injective (in characteristic not $2$ the canonical map is an isomorphism of $M_n(k)$ modules).