Galois Groups and Fundamental Groups (Cambridge Studies in Advanced Mathematics, Series Number 117) 1st Edition by Tamás Szamuely (PDF)

Description

Ever since the concepts of Galois groups in algebra and fundamental groups in topology emerged during the nineteenth century, mathematicians have known of the strong analogies between the two concepts. This book presents the connection starting at an elementary level, showing how the judicious use of algebraic geometry gives access to the powerful interplay between algebra and topology that underpins much modern research in geometry and number theory. Assuming as little technical background as possible, the book starts with basic algebraic and topological concepts, but already presented from the modern viewpoint advocated by Grothendieck. This enables a systematic yet accessible development of the theories of fundamental groups of algebraic curves, fundamental groups of schemes, and Tannakian fundamental groups. The connection between fundamental groups and linear differential equations is also developed at increasing levels of generality. Key applications and recent results, for example on the inverse Galois problem, are given throughout.

User’s Reviews

Editorial Reviews: Review “The book is well written and contains much information about the etale fundamental group. There are exercises in every chapter. On the whole, the book is useful for mathematicians and graduate students looking for one place where they can find information about the etale fundamental group and the related Nori fundamental group scheme.” Swaminathan Subramanian, Mathematical Reviews Book Description Assuming little technical background, the author presents the strong analogies between these two concepts starting at an elementary level. About the Author Tamás Szamuely is a Senior Research Fellow in the Alfréd Rényi Institute of Mathematics at the Hungarian Academy of Sciences, Budapest. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book answered a lot of my needs- the coverage on the fundamental groups of schemes made my senior thesis a lot better- the exposition helped my understand the ideas clearly, which translated to my writing. I recommend this to any mathematician looking for a fun book (we do have a characteristic definition of fun). I appreciate the authors writing style- it does not have pretentious/know-it-all tone.

⭐”Everyone” who has taken a course covering Galois Theory of Fields and a course covering Fundamental Groups of Topological Spaces (that is to say, strong undergraduate students and beginning graduate students in mathematics) recognizes that the correspondence between Galois extensions and subgroups of the absolute Galois group is “the same thing” as the correspondence between covering spaces and subgroups of the fundamental group. This is not an accident, but rather the result of a rich and deep underlying theory developed by Grothendieck.To begin, the prerequisites for this book are familiarity with introductory courses in field theory, algebraic topology, and complex functions. Although the exploration of this connection between algebra and topology eventually leads deep into the land of algebraic geometry, no background is assumed.The first two chapters give a rapid but solid recap of Galois theories of fields and topological spaces, with the bonus of recasting the “main theorem” of both in a Grothendieckian view as preparation for later chapters: Galois theory of fields in terms of étale algebras, and Galois theory of topological spaces in terms of locally constant sheaves. The rest of the book slowly builds the underlying theory, in several stages. Chapter three hints at the algebraic geometry view of Galois theory by connecting the covering spaces of Riemann surfaces with étale algebras. Chapter four generalizes the story of chapter three by bringing us into algebraic geometry land proper by investigating the Galois theory of varieties, and then leads us into the deeper waters of Galois theory of schemes studied in Chapter five.The book is masterfully written: even students who are not fully comfortable with Galois theory of fields and topological spaces will have no trouble with this book, as it is written in a very friendly manner with many examples with which to ground the reader. The choice to slowly build up to the theory of schemes is very nice: most students who have seen Galois groups and Fundamental groups have not seen any algebraic geometry, and yet it is not even necessary to take an algebraic geometry course while reading this book (of course, it certainly won’t hurt to do so). At first glance, the last chapter may seem incredibly intimidating, but Szamuely has written the book in such a way that by the time the reader gets there, it will seem just as straightforward as the first few chapters.

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