Exercise 8, § 5

Let $A$ be a Noetherian local ring, $m$ its maximal ideal and $k = A/m$ its residue field. Let $K$ be an extension of $k$; show that there exists a local homomorphism from $A$ to a Noetherian local ring $B$ such that $B/mB$ is isomorphic to $K$ and $B$ is a flat $A$-module. (Consider first the case where $K = k(t)$, distinguishing two cases according to whether $t$ is algebraic or transcendental over $k$; consider next a family $(K_\lambda)$ of subfields of $K$ containing $k$, which is well ordered by inclusion and such that, if $K_\lambda$ has a predecessor $K_\mu$, $K_\lambda = K_\mu(t_\mu)$ for some $t_\mu \in K_\lambda$. Finally, apply Exercise 7.)