  Fusion categories are fundamental objects in quantum algebra, but their
definition is narrow in some respects. By definition a fusion category must be
k-linear for some field k, and every simple object V is strongly simple,
meaning that (V) = k. We prove that linearity follows automatically from
semisimplicity: Every connected, finite, semisimple, rigid, monoidal category
\C is k-linear and finite-dimensional for some field k. Barring inseparable
extensions, such a category becomes a multifusion category after passing to an
algebraic extension of k.
  The proof depends on a result in Galois theory of independent interest,
namely a finiteness theorem for abstract composita.
