Maximal fields disjoint from certain sets

Suppose that C is an algebraically closed field and that Q is a subfield of C. If S is a nonempty subset of C disjoint from Q, it follows from an application of Zorn's lemma that there is a subfield k of C which is maximal with respect to the properties that QCk and k and S are disjoint. The problem is to describe the field extension C/k. When S consists of a single element this has been done by Quigley [4, Theorems 1, 2 and 3]. In this note we shall give several theorems which describe C/k when S consists of exactly two elements. When S contains more than two elements, some of the arguments used in the proof of Theorem 2 fail. The first theorem holds when S is any finite (nonempty) subset of C disjoint from Q. I t generalizes one of Quigley's results [4, Lemma 1].