Associated to a finite, separable field extension $L/K$, there is a natural nondegenerate bilinear form, the trace form, defined by $$\langle x,y \rangle := \mathrm{Tr}_{L/K}(xy)$$

Now, given a finite dimensional $K$-vector space $V$ with a nondegenerate bilinear form $\langle,\rangle$ what are some interesting/useful necessary or sufficient conditions for $\langle, \rangle$ to be equal to the bilinear form associated to a finite separable field extension $L/K$? Or more generally to some algebra $A/K$?