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I'm currently working on my undergraduate dissertation. I'm working on covering sapces of Riemann surfaces so my supervisor asked me to read the book I mention in the title: "Algébre et Théories Galoisiennes" by Régine and Adrien Douady. The final scope is to prove the following theorem in the book:

Let $ B$ be a Rieamann surface, $v_B $ the category of ramified analytic coverings of $B$ and $e_B $ the category of étale algebras over $M(B)$ (the field of meromorphic functions on $B$). Then the functor $M: X\mapsto M(X) $ from $v_B $ to $e_B $ is an antiequivalence of categories.

The problem is that the suggested literature is in french so it took three times the average time to read a page. Besides, the topic is totally new for me so I'm not making any progress. I'm aware of the fact that probably there is not a translation of that document so what I'm looking for is a text in English which deals with this theorem.

If some of you could suggest some literature I would be very grateful. Thank you all in advance.

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    $\begingroup$ I'm not sure what level of expertise or approach to Riemann surfaces you're looking for, but a fantastic book is Szamuely's Galois Groups and Fundamental Groups. He discusses several very different perspectives to the phenomenon of coverings. Chapter 3 is Riemann surfaces, but you can also see comparisons with topological spaces, field extensions, algebraic varieties, and schemes. $\endgroup$
    – PrimeRibeyeDeal
    Commented Nov 8, 2018 at 1:10
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    $\begingroup$ In particular, look at Theorem 3.3.7 on p82. $\endgroup$
    – PrimeRibeyeDeal
    Commented Nov 8, 2018 at 1:27
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    $\begingroup$ Taking the chance to learn to read mathematical French more fluently when you (a) have a specific text in which you are interested and (b) have the time (it may not seem like it, but, as an undergraduate, you probably have more time than you probably will for the rest of your academic career!) is initially painful but very very very much worth it in the long run. $\endgroup$
    – LSpice
    Commented Nov 8, 2018 at 2:18
  • $\begingroup$ There is no exact English equivalent. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 8, 2018 at 2:50
  • $\begingroup$ @LSpice I agree with you that this si a good change since I will find much more mathematical texts to read in the future, but I think that having an english book supporting me while reading Douady would be nice. $\endgroup$
    – Natalio
    Commented Nov 8, 2018 at 14:25

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O. Forster "Lectures on Riemann Surfaces" (Springer) is a good starting point before taking on T. Szamuely "Galois Groups and Fundamental Groups". After all, as Szamuely writes on page 65 at the beginning of Chapter 3, parts of his exposition in this chapter were inspired by Douady and Forster.

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  • $\begingroup$ Girondo--González-Diez' "Introduction to Compact Riemann Surfaces and Dessins d’Enfants" might be useful as well. $\endgroup$
    – archipelago
    Commented Nov 9, 2018 at 13:21
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An English translation of the book by the Douadys is scheduled to come out with Springer early in 2020.

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