We say that a deterministic finite automaton (DFA) $\A$ is {\em
composite\/} if there are DFAs $\A_1,\ldots,\A_t$ such that $L(\A) =
\bigcap_{i=1}^t L(\A_i)$ and the index of every $\A_i$ is strictly
smaller than the index of $\A$. Otherwise, $\A$ is {\em prime}. We
study the problem of deciding whether a given DFA is composite, the
number of DFAs required in a decomposition, methods to prove
primality, and structural properties of DFAs that make the problem
simpler or are retained in a decomposition.