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I have been studying galois theory on my own and find it very fascinating. I have gone through Ian Stewarts book: 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.

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I think inverse Galois problems and Galois embedding problems could be interesting subjects to continue. Just google these phrases to find reading materials that suits you. You can download my book on Galois embedding problem at researchgate: (…). – Vahid Shirbisheh Apr 3 '13 at 9:19
Look at the list of books given in In addition to that, also the book Field Arithmetic by Fried and Jarden contains quite a bit of (infinite) Galois theory. Furthermore, Algebraic Patching by Jarden is mostly about a relatively recent technique in Galois theory. – Peter Mueller Apr 3 '13 at 13:24
You could go to class field theory, to algebraic geometry, to combinatorics... it depends on your tastes, really. If you could give more insight about what you like, you would have a higher chance to get good references. – Julien Puydt Apr 4 '13 at 8:36
The book Borceux, F. and Janelidze, G. Galois theories, Cambridge Studies in Advanced Mathematics, Volume~72, (2001) does not seem to be mentioned on the wiki sitea, and does give a more general view of Galois Theory, derived from Magid, Grothendieck, .... and including rings. and algebras. The general theory involves Galois Groupoids. The Preface has a good discussion of the context of the generalisations. The details though are quite advanced. – Ronnie Brown Apr 4 '13 at 10:12
If you liked Stewart's Galois Theory book, you might like Stewart and Tall's Algebraic Number Theory (I prefer the 1st edition if you can find a used copy, but probably the current edition is good as well) – Yemon Choi Jul 19 '13 at 0:57

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.

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For more advanced material, you might like to try Askold Khovanskii: Galois Theory, Coverings, and Riemann Surfaces.

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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.

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A good introductory textbooks in Galios theory are (Foundations of Galois Theory, by M.M. Postnikov) and (A Classical Introduction to Galois Theory, by S.C. Newman). The book by Borceux and Janelidze, mentioned in comments, 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: (Introduction to Finite Fields and their Applications).

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