  There exist several theorems which state that when a matroid is representable
over distinct fields F_1,...,F_k, it is also representable over other fields.
We prove a theorem, the Lift Theorem, that implies many of these results.
  First, parts of Whittle's characterization of representations of ternary
matroids follow from our theorem. Second, we prove the following theorem by
Vertigan: if a matroid is representable over both GF(4) and GF(5), then it is
representable over the real numbers by a matrix such that the absolute value of
the determinant of every nonsingular square submatrix is a power of the golden
ratio. Third, we give a characterization of the 3-connected matroids having at
least two inequivalent representations over GF(5). We show that these are
representable over the complex numbers.
  Additionally we provide an algebraic construction that, for any set of fields
F_1,...,F_k, gives the best possible result that can be proven using the Lift
Theorem.
