Exercise 111.5.7. Let $(A, {\mathfrak m}, k)$ be a local ring and let $k'/k$ be a finite field extension. Show there exists a flat, local map of local rings $A \to B$ such that ${\mathfrak m}_ B = {\mathfrak m} B$ and $B/{\mathfrak m} B$ is isomorphic to $k'$ as $k$-algebra. (Hint: first do the case where $k \subset k'$ is generated by a single element.)
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