For a field extension (Formula presented.) we consider maps that are quadratic over (Formula presented.) but whose polarisation is only bilinear over (Formula presented.). Our main result is that all such are automatically quadratic forms over (Formula presented.) in the usual sense if and only if (Formula presented.) 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.