On Kaplansky's Embedding Theorem

Let $R$ be a complete equicharacteristic noetherian local domain with an algebraically closed residue field $k$. Let $\nu$ be a zero dimensional valuation of rank one centered in $R$ with value group $\mathrm{\Phi }$. We show that there is a valuation preserving embedding of $R$ in the ring $k[[t^{{\mathrm{\Phi }}_{\ge 0}}]]$ of Hahn series. The method relies on torific local uniformization for rational valuations with finitely generated semigroup and the approximation of rational valuations by semivaluations with finitely generated semigroup.