Books on advanced galois theory I have been studying galois theory on my own and find it very fascinating. I have gone through Ian Stewarts book: http://www.amazon.co.uk/Galois-Theory-Third-Chapman-Mathematics/dp/1584883936. I am not sure which book to study from next but I would like to learn about something along the lines of algebraic closures of Q and other related topics. 
Related to this, I would also like to learn more finite field theory. I would like books that are not too technical and provide some context since I am doing a self study.
 A: For more advanced material, you might like to try Askold Khovanskii: Galois Theory, Coverings, and Riemann Surfaces.
A: For finite fields, there is Lidl and Niederreiter, Finite Fields, which is Volume 20 in the Encyclopedia of Mathematics and its Applications. There are also a couple of conference proceedings: Finite Fields and Applications, the proceedings of the 3rd international conference on finite fields and applications, edited by Cohen and Niederreiter, and Finite Fields: Theory, Applications, and Algorithms, the proceedings of the 4th international conference, edited by Mullin and Mullen. There may be several other books in this series of conference proceedings, I've just mentioned the two I have in my office. 
A: The book, Algèbre et théories galoisiennes, by Adrien and Régine Douady, discusses Galois theory vs. the topological theory of coverings, especially in the context of Riemann surfaces. It concludes by an introduction to the theory of dessins d'enfants.
A: A good introductory textbooks in Galios theory are http://www.amazon.com/Foundations-Galois-Theory-Dover-Mathematics/dp/0486435180 (Foundations of Galois Theory, by M.M. Postnikov) and http://onlinelibrary.wiley.com/book/10.1002/9781118336816 (A Classical Introduction to Galois Theory, by S.C. Newman). The book by Borceux and Janelidze, mentioned in comments, http://www.amazon.com/Theories-Cambridge-Studies-Advanced-Mathematics/dp/0521070414 is a comprehensive textbook with modern perspective. However its later chapters are much harder to read than the earlier ones.
Lidl and Niederreiter's monograph on finite fields, recommended by Gerry Myerson,
has a textbook version: http://www.amazon.com/Introduction-Finite-Fields-their-Applications/dp/0521460948 (Introduction to Finite Fields and their Applications).
