  This paper is devoted to the study of reducing subspaces for multiplication
operator $M_\phi$ on the Dirichlet space with symbol of finite Blaschke
product. The reducing subspaces of $M_\phi$ on the Dirichlet space and Bergman
space are related. Our strategy is to use local inverses and Riemann surface to
study the reducing subspaces of $M_\phi$ on the Bergman space, and we discover
a new way to study the Riemann surface for $\phi^{-1}\circ\phi$. By this means,
we determine the reducing subspaces of $M_\phi$ on the Dirichlet space when the
order of $\phi$ is $5$; $6$; $7$ and answer some questions of
Douglas-Putinar-Wang \cite{DPW12}.
