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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: 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.

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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: (researchgate.net/publication/…). –  Vahid Shirbisheh Apr 3 '13 at 9:19
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Look at the list of books given in en.wikipedia.org/wiki/Inverse_Galois_theory 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
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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
    
Hey,I posted the topic. The problem is, I have no idea of further fields and would really like to try them all. So any suggestions at my level would be very welcome indeed. –  user32779 Apr 4 '13 at 9:19
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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

2 Answers 2

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

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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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