In his review of the first edition of Cox's Galois Theory, Darren Glass described it as "a fantastic book." Eight years later, we have a second edition, and it's still a fantastic book.
Books on Galois theory seem to come in two styles. There are the "short form" books, which assume most of the abstract algebra background and make a beeline for the Galois correspondence, with minimal discussion of side issues or examples. The great classic of short form Galois theory are the notes from Emil Artin's Notre Dame lectures, now available both as a little volume from Dover and in the collection Exposition by Emil Artin.
Long form Galois theory books are less common but much richer. Cox's Galois Theory must now be considered the standard. It has very broad coverage of field theory, Galois theory, and many related topics. Examples are explored in depth, as is the historical background. There is a big chapter on the explicit computation of Galois groups, a discussion of solvable subgroups of permutation groups, and an analysis of the division polynomials of the lemniscate that can serve as an introduction to elliptic functions in the classical style.
Additions to the second edition include a complete (i.e., not restricted to characteristic zero) treatment of the computation of the Galois groups of degree four polynomials. A very nice addition is a list of suggested student projects.
Galois Theory is readable, interesting, and thorough. An excellent book.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.