# The trace of $F|K$ is non degenerate if and only if the trace of $E\otimes_{K}F|E$ is nondegenerate

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.

• Hm. Are you sure you mean $Tr_{E\otimes_KF\mid K}$? Oct 14, 2015 at 7:48
• My answer is about $Tr_{E\otimes_KF\mid E}$. If the characteristic of $K$ is zero, for example, the trace form over $K$ of $E\otimes_KF$ is nondegenerate iff $E\otimes_KF$ is a semisimple $K$-algebra, and your hypotheses are not enough to guarantee that. Oct 14, 2015 at 8:10
• the usualone, if the first is made up of elements $f_i$, the second one is just the one made up with the corresponding $1\otimes f_i$. Oct 14, 2015 at 8:27
• I suggest that you try to prove that $\{1\otimes f_i\}$ is a basis of $E\otimes_KF$ as an $E$-vector space. That $\{f_i\}$ is not a basis of $F$ as an $E$-vector space has nothing to do with this (and in fact does not even make sense, as $F$ is not an $E$-vector space in any sensible way :-) ) Oct 14, 2015 at 8:37
• The $L$ is probably a typo. Oct 14, 2015 at 8:40