Finding little hyperplanes in bigger ones

AbstractIf L is a field extension of K and V is an L-vector space, when is it possible to find in a K-hyperplane in V some L-hyperplane? If the L-dimension of V is at least 2, then this is possible for every K-hyperplane in V iff L is a finite degree extension of K. If L is infinite over K and if V is large enough, there will always be K-hyperplanes in V which contain no L-subspace of countable codimension. A study of K-linear functionals on V which factor through L is also given. Some explicit examples of K-hyperplanes which contain no L-hyperplanes are provided