  In this paper we consider a class of Einstein warped product semi-Riemannian
manifolds $\widehat{M} = M^{n}\times_{f}N^{m}$ with $n\geq 3$ and $m\geq 2$.
For $\widehat{M}$ with compact base and Ricci-flat fiber, we prove that
$\widehat{M}$ is simply a Riemannian product space. Then, when the base $M$ is
conformal to a pseudo-Euclidean space which is invariant under the action of a
$(n-1)$-dimensional translation group, we classify all such spaces.
Furthermore, we get new examples of complete Einstein warped products
Riemannian manifolds.
