Exercise 5, Appendix 0

Let A be a local ring, with residue field $\kappa(A)$. Let B be an inflation of A, with residue field $\kappa(B)$. a) If $\kappa(B) = \kappa(A)$, one has $A = B$. b) If $\kappa(B)$ is algebraic over $\kappa(A)$, the A-algebra B is integral. c) If $\kappa(B)$ is of finite degree over $\kappa(A)$, B is a free A-module whose rank is the degree of $\kappa(B)$ over $\kappa(A)$.