  In this paper we consider connections between Ricci solitons and Einstein
metrics on homogeneous spaces. We show that a semi-algebraic Ricci soliton
admits an Einstein one-dimensional extension if the soliton derivation can be
chosen to be normal. Using our previous work on warped product Einstein
metrics, we show that every normal semi-algebraic Ricci soliton also admits a
$k$-dimensional Einstein extension for any $k\geq 2$. We also prove converse
theorems for these constructions and some geometric and topological structure
results for homogeneous warped product Einstein metrics. In the appendix we
give an alternative approach to semi-algebraic Ricci solitons which naturally
leads to a definition of semi-algebraic Ricci solitons in the non-homogeneous
setting.
