  For a field extension $L/K$ we consider maps that are quadratic over $L$ but
whose polarisation is only bilinear over $K$. Our main result is that all such
are automatically quadratic forms over $L$ in the usual sense if and only if
$L/K$ is formally unramified. In particular, this shows that over finite and
number fields, one of the axioms in the standard definition of quadratic forms
is superfluous.
