The u-invariant of a field is defined to be the maximal dimension (number of variables) of a quadratic form which has no nontrivial zeros. Although there are some expectations for what u-invariants should be of most "naturally occuring" fields, these invariants are unknown in a great number of situations. For example, if $F$ is a nonreal number field, it is known that $u(F) = 4$, and it is expected that the u-invariant of the rational function field $F(t)$ should be $8$. At this point, however, there is no known bound for $u(F(t))$ (and no proof it is even finite). Important progress on this type of problem was obtained by Parimala and Suresh late last year, who showed that the u-invariant of a rational function field $F(t)$ is $8$ when $F$ is $p$-adic ($p$ odd). In this talk I will describe joint work with David Harbater and Julia Hartmann in which we give an independent proof and a generalization of this result using the method of ``field patching."