An infinite sequence of real random variables (ξ1,ξ2,…) is said to be rotatable if every finite subsequence (ξ1,…,ξn) has a spherically symmetric distribution. A classical theorem of David Freedman says that (ξ1,ξ2,…) is rotatable if and only if ξj=σηj for all j, where (η1,η2,…) is a sequence of independent standard Gaussian random variables and σ is an independent nonnegative random variable. We establish the analogue of Freedman's result for sequences of random variables taking values in arbitrary locally compact, nondiscrete fields other than the field of real numbers or the field of complex numbers. This is joint work with Daniel Raban, a Berkeley undergraduate.