The book “Galois Groups and Fundamental Groups” by Tamás Szamuely is an expository text which was written so as to give an overview of the many different contexts in which Galois groups and fundamental groups occur. It uses the overarching theme of exploring the relationship between these two types of groups to introduce many classical and modern concepts which in my opinion both gives it a coherent storyline and makes it enjoyable to read. I read chapters 1,2,3 and parts of chapter 5 and solved many of the exercises provided at the end of each chapter.

Chapter 1 gives a review of the Galois theory of fields with a focus on the more advanced/abstract parts such as infinitary Galois theory and Grothendieck’s version of the fundamental theorem of Galois theory. This chapter is suitable for anyone who already has some experience with basic field theory but is probably not ideal as a textbook for learning Galois theory from scratch. Namely, most of the proofs of the more basic results are omitted and only the first approximately 10 pages are dedicated to finitary Galois theory.

Chapter 2 discusses covering spaces and the fundamental group. I believe once again that this chapter is most likely not particularly useful as a first introduction into covering theory. In particular, I find the way in which the universal cover is presented to be rather unenlightening if one is not already familiar with this construction. Moreover, there are no pictures contained in the book which makes it hard to visualize the new concepts if one does not already have some kind of mental image. On the other hand I highly recommend reading this if you have a bit of a background in covering theory: Much of the theory presented has a rather category theoretical spin to it which offers an alternative view on the more classical definitions. Moreover, Tamás Szamuely takes care to formulate the results and definitions so as to highlight the structural similarity with the previous chapter whenever possible. I skipped sections 2.6 and 2.7 since I was not planning on learning about differential equations at the time though the introduction to sheaves preceding these two sections should not be skipped as it will become essential later on.

Chapter 3 then links chapters 1 and 2 by discussing the theory of Riemann surfaces. On the one hand, it explores the topological properties of Riemann surfaces and how they relate to covering theory and on the other hand, the meromorphic function field is introduced which allows one to apply the results from chapter 1. Thereafter, some applications are discussed and in particular the absolute Galois group of C(t) is described. All in all I found this chapter to be very elegant and interesting and at the same time an accessible introduction into the theory of Riemann surfaces, assuming only basic complex analysis.

Finally, chapter 5 introduces schemes and finite étale covers of schemes. I ignored the warning at the beginning of this chapter and read it with no prior knowledge on schemes and modern algebraic geometry. I found that it was nevertheless possible to follow the arguments but as the introduction to schemes is rather short, proceeding in this way will most likely leave you with only the intuition or general idea of most of the proofs while more knowledge must be obtained from a different source such as “The rising sea” or EGA in order to get a better understanding.

All in all, I found this book to be quite entertaining and informative and in a pinch it can also serve as a reference for both basic algebraic geometry and topology. I decided to skip chapter 4, which discusses algebraic curves, as I am mainly interested in schemes anyways and half way through this chapter I was kind of put off by the sheer amount of definitions appearing (many of the concepts will have a lot of adjectives attached to them). However, in a way this also provides a motivation for the introduction of schemes as the greater generality seems to make the concepts clearer. The exercises at the end of each chapter are very helpful and I think that solving a significant portion of them is indispensable for understanding the book.

Note however that some of them do contain errors: A collection of all currently known errata can be found on the author’s homepage.