Flat extension of local rings with a specified extension of residue field

Question

Let $(R, \mathfrak m_R, k)$ be a Noetherian local ring and $K$ be a field containing $k$.

Then is it true that there is a Noetherian local ring $(S, \mathfrak m_S)$ and a flat ring homomorphism $f: R\to S$ such that $f(\mathfrak m_R)S=\mathfrak m_S$ and $S/\mathfrak m_S\cong K$ ?

Answer 1

Yes. You can see a proof in Bourbaki, Algèbre Commutative, chapitre IX, Appendice, n.2, Corollaire du Théorème 1.