I'm supervising an undergraduate project on Galois theory and covering spaces. I want to have him read about the fact that from a branched cover of a Riemann surface you get an extension of its field of meromorphic functions, and the Galois groups are the same, but I'm having trouble finding a good book. Fulton's "Algebraic Topology" is OK but rushes through this point. Forster's "Lectures on Riemann Surfaces" looks good but I'd rather not make him learn sheaves. Any recommndations?

7$\begingroup$ What about "Algebre et théories galoisiennes" of A. Douady and R. Douady (if you can read french)? $\endgroup$– HenriFeb 17, 2012 at 18:00

2$\begingroup$ McKean and Moll's book Elliptic Curves might be a bit elementary but I like their discussion of this a lot. $\endgroup$– j.c.Feb 17, 2012 at 18:05

$\begingroup$ @Henri: that would have been my suggestion, too. $\endgroup$– Franz LemmermeyerFeb 17, 2012 at 18:06

$\begingroup$ Douady and Douady is appealing to me, but it's probably too sophisticated for my student  they define the field of meromorphic functions as a projective limit, for example. McKean and Moll is more the right style, although I'm having trouble finding where they address the fact that I asked about. $\endgroup$– Nick AddingtonFeb 18, 2012 at 0:09

$\begingroup$ Simon Donaldson's new book "Riemann Surfaces" looks very nice if I could scare up a copy... $\endgroup$– Nick AddingtonFeb 18, 2012 at 0:20
10 Answers
There is a chapter on Riemann surfaces in Tamās Szamuely's book "Galois Groups and Fundamental Groups", which contains the facts that you are looking for.

$\begingroup$ Thanks, although judging by the table of contents it may be too sheafy for an undergrad. Anyway it seems hard to come by  I can't find it in the library or online. $\endgroup$ Feb 17, 2012 at 23:39

$\begingroup$ The British library has it (Link 1 below), and I think you can order it via Imperial´s main library. www3.imperial.ac.uk/library/find/documentdelivery/faqs Link 1) explore.bl.uk/primo_library/libweb/action/… $\endgroup$– James OFeb 20, 2012 at 13:19
I think V.I.Arnold's lectures "Abel's Theorem in Problems and Solutions" may be a great supplementary reading. The book is basic but beautiful.

1$\begingroup$ I'm an undergrad who recently started working through this myself. $\endgroup$– mmmFeb 17, 2012 at 22:01

2$\begingroup$ What a lovely book. Probably too elementary for this student, but I hope I'll find an excuse to use it someday. $\endgroup$ Feb 17, 2012 at 23:09

1$\begingroup$ There is an apparently different tranaslation of the russina lectures which can be downloaded for free from the website of Sujit Nair. I hope it is allright to point this out here. $\endgroup$ Oct 19, 2012 at 8:54
Here are two sources accesible to an undergraduate
1. Michio Kuga: Galois' Dream. Group theory and differential equations, Birkhauser.
It's written with an undergraduate in mind that is not familiar with the fundamental group and/or covering spaces. He does not cover branched covers though.
2. F. Kirwan: Complex Algebraic Curves, London Math. Soc., Student Texts, vol. 23.

$\begingroup$ Thanks, I will check out the Kuga reference. Kirwan doesn't mention Galois groups at all. $\endgroup$ Feb 17, 2012 at 22:29
I like a lot:
Algebraic Curves and Riemann Surfaces by Rick Miranda,
published by the AMS. I think it's very suitable for undergraduates.

2$\begingroup$ This doesn't seem to mention anything the Galois group of a branched cover. $\endgroup$ Feb 17, 2012 at 22:25

$\begingroup$ In III.2 it describes cyclic covers of the line, III.3 is about group actions on Riemann surfaces, III.4 treats monodromy. All this stuff seems to me closely related to your question. $\endgroup$– ritaFeb 18, 2012 at 10:12
Try Askold Khovanskii: Galois Theory, Coverings and Riemann Surfaces.

$\begingroup$ A new book by the same author just came out: Topological Galois Theory. $\endgroup$ Nov 3, 2014 at 14:34
Hi Nick,
Groups as Galois groups by Helmut Völken is a very nice book that I think is suitable for a good undergrad and might have the level you are looking for. I think chapters 4 and 5 are the places where your student should check first, and I think they don't require previous chapters to follow what is there.
Also, Inverse Galois theory by Malle and Matzat is great to see some applications of what he is learning is his projectmainly chapter 1. This one needs more background than the above, so I'm just recommending this one after he has learned the material in the other one.
Approaching the problem from a slightly different position, you could point your student towards a Masters' thesis:
M. A. D. Robalo, 2009, Galois Theory towards Dessins d’Enfants , Master’s thesis, Instituto Superior Technico, Lisboa.
There are some more or small errors, and the aim is slightly different, so the task might then be to rewrite that (slightly too SGA1 based perhaps), to check for errors, adapting it towards the aims that you have in mind and bringing in more Riemann surface stuff.
I am not sure but the, "introduction to Compact Riemann Surfaces and Dessins d'Enfants" of Ernesto Girondo & Gabino GonzálezDiez. Could be useful
The first sections of the following two papers contain background material on covering spaces and Galois theory.
Joe Harris Galois groups of enumerative problems Duke Math. J. Volume 46, Number 4 (1979), 685724.
William Fulton Hurwitz Schemes and Irreducibility of Moduli of Algebraic Curves. The Annals of Mathematics, Second Series, Vol. 90, No. 3 (Nov., 1969), pp. 542575
Klaus Lamotke: Riemmansche Flächen, Springer, 2009. http://books.google.pt/books?id=lfrtVbRPtn4C Very strong on algebraic aspects. (Don't know of anything alike in english, yet.)