  We establish an analytic Hasse principle for linear spaces of affine
dimension m on a complete intersection over an algebraic field extension K of
Q. The number of variables required to do this is no larger than what is known
for the analogous problem over Q. As an application we show that any smooth
hypersurface over K whose dimension is large enough in terms of the degree is
K-unirational, provided that either the degree is odd or K is totally
imaginary.
