Introduction to the Galois Correspondence

This monograph is a self-contained text book which assumes only that the student has a certain level of mathematical sophistication. The introductory chapter cover such topics as Sylow p -subgroups, solvable groups, and the structures of finite, abelian groups, thus providing the student with a firm foundation for the study of the Galois correspondence. Appendices covering group actions, the Sylow theorems, generators and relations, Euclidean domains, principal ideal domains, unique factorization domains and linear algebra have been added to provide a more in-depth coverage of abstract algebra. The Galois correspondence itself is presented with many well-constructed, concrete examples and exercises of varying degrees of difficulty. Some of the diverse applications of the Galois correspondence are presented, including the Fundamental Theorem of Algebra, the unsolvability of the general quintic, classical constructibility problems, Wedderburn's theorem on finite division rings, and a special case of Dirichlet's theorem (concerning the existence of infinitely many prime integers of the form nk + 1). This self-contained text, with its clear exposition and abundance of examples, exercises and applications is an excellent guide for hte student who is seeking an understanding of the power and the elegance of the Galois correspondance in mathematics.