It is not uncommon to describe interesting classes of field extensions by declaring that an extension $L|K$ belongs to that class if some type of problem with $K$-coefficiens has a property over $L$ if and only if it has the same property over $K$. I wonder about the following variant:

Question A: For which field extensions $L|K$ the following is true?: Given finite dimensional $K$-vector spaces $U,V,W$, a $K$-bilinear form $\beta_K:U\times V\to W$ is surjective if and only if the corresponding $L$-bilinear form $\beta_L$ obtained by scalar extension is surjective.

Already in characteristic zero an answer to that would be nice. Also, I wonder for which $L|K$ surjectivity of $\beta_K$ implies or is implied by surjectivity of $\beta_L$.

One can take the geometric point of view: A bilinear form induces a map between associated projective spaces and one asks here for surjectivity of these maps on $K$ or $L$-rational points.

It is not hard to show that if $L$ is the reals or the $p$-adics, then surjectivity of a bilinear form over the rationals implies surjectivity with $L$-coefficients (the argument really uses both, density *and* local compactness). This is the setting in which the problem originally arised. I was also asking the following

Question B: Given a bilinear form $\beta$ between finite dimensional $\mathbb Q$-vector spaces, is it true that $\beta$ is surjective if and only if for all primes $p$ (including $p=\infty$) the induced $\mathbb Q_p$-bilinear form is surjective.

The answer to that is negative, see Poonen's explicit example below.

inducesa map on the tensor product, but they are not thesame. In particular, the latter can be surjective even if the former is not. $\endgroup$ – Bjorn Poonen May 19 '10 at 1:50