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.
Categories: math.RA