This is not an answer per se, but: I have recently been thinking about the Davenport-Cassels Theorem and found a generalization to the context of "normed rings" which also recovers the Cassels-Pfister Theorem as a special case. (This also makes precise KConrad's assertion in the comments to his answer that the latter result is a function field analogue of the former one.) See
http://www.math.uga.edu/~pete/CDforms.pdf
for a preliminary writeup. I think those who are interested in this question may also be interested in this paper and have useful input to give: comments are most welcome. (Note also that, if this answer is an advertisement for my work on MO, the favor is returned in the paper: Bjorn's answer is cited there.)
