Let $F|K$ be a finite field extension. I want to prove that if $E|K$ is an algebraic extension, then the bilinear form $$ F\times F\rightarrow K, (x,y)\mapsto Tr_{F|K}(xy) $$ is nondegenerate if and only if $$ E\otimes_{K} F\times E\otimes_{K} F\rightarrow E, (x,y)\mapsto Tr_{E\otimes_{K}F|E}(xy) $$ is nondegenerate.
I don't know where to start from.
Non-degeneracy of a bilinear form is equivalent to having the matrix of the form with respect to a basis have non-zero determinant.
Show that your two bilinear forms have matrices with respect to appropriate bases of their underlying vector spaces which have the same determinant — in fact, the two matrices can be chosed to be the same matrix!