  Shimura proved that each principally polarized abelian variety over
$\mathbf{C}$ admits a unique factorization into irreducible principally
polarized abelian varieties. We give an exposition of his result, and
generalize to an arbitrary ground field $k$. If $k$ is separably closed, the
irreducible factors are in bijection with the irreducible components of a theta
divisor over $k$ giving rise to the polarization.
