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.