Corollary

Let A be a local ring and K an extension of its residue field $\kappa_{A}$. Then there exist a local ring B and an inflation $A \to B$ such that the $\kappa_{A}$-algebra $\kappa_{B}$ is isomorphic to K.

In fact, the homomorphism $\kappa_{A} \to K$ is an inflation (Example 1). Applying Theorem 1 with $A' = \kappa_{A}$ and $B' = K$, one obtains the existence of an inflation B of A and of an A-isomorphism of $B/m_{A}B$ onto K, whence the corollary.