Maximal subfields of algebraically closed fields

AbstractLet Kbe an algebraically closed field of characteristic zero, and Sa nonempty subset of Ksuch that S Q= ? and card S< card K, where Qis the field of rational numbers. By Zorn's Lemma, there exist subfields Fof Kwhich are maximal with respect to the property of being disjoint from S. This paper examines such subfields and investigates the Galois group Gal K/F along with the lattice of intermediate subfields.