Tell me more ×  This is just a reference request on a relatively
elementary level (for which I apologize in advance), but every
time I bump into this question I suspect I'm missing the
"correct" conceptual setting. In the simplest case, one is given
two vector spaces $V, W$ over a field of characteristic 0, each
endowed with a symmetric bilinear form. Then the tensor product
$V \otimes W$ inherits an obvious symmetric bilinear form. A
natural result is that nondegeneracy of the given forms implies
nondegeneracy of the new form, though the proof seems to require
somewhat messy manipulation of bases and indices. Even if the
vector spaces are infinite dimensional, the same principle seems
valid. Then there is the possibility of working over a field of
prime characteristic, as well as passing to free modules over
commutative rings, etc.  Where in the textbook literature can
one find the most definitive treatment of nondegeneracy of
symmetric bilinear forms on tensor products? (Preferably with
few indices to keep