  Let $K$ be a field, let $E$ be a finite directed graph, and let $L_K(E)$ be
the Leavitt path algebra of $E$ over $K$. We show that for a prime ideal $P$ in
$L_K(E)$, the following are equivalent: \begin{enumerate} \item $P$ is
primitive; \item $P$ is rational; \item $P$ is locally closed in ${\rm
Spec}(L_K(E))$. \end{enumerate} We show that the prime spectrum ${\rm
Spec}(L_K(E))$ decomposes into a finite disjoint union of subsets, each of
which is homeomorphic to ${\rm Spec}(K)$ or to ${\rm Spec}(K[x,x^{-1}])$. In
the case that $K$ is infinite, we show that $L_K(E)$ has a rational
$K^{\times}$-action, and that the indicated decomposition of ${\rm
Spec}(L_K(E))$ is induced by this action.
