Exercise 4, § 3

a) Let k be a field and K an extension of k. If K is separable over k, then K is a smooth k-algebra. (If K is an extension of finite type over k, use Prop. 1 of § 3, No. 2. In the general case, write K as a union of extensions of finite type over k and use the preceding exercise.) b) Let A be a separated and complete local ring containing a field k. Using a), prove that A possesses a field of representatives. If the residue field of A is a separable extension of k, then A possesses a field of representatives containing k.