12 The Weak Mordell-Weil Theorem 123 12.1 Kummer Theory of Number Fields . . . . . . . . . . . . . . . . . . . 123 12.2 Proof of the Weak Mordell-Weil Theorem . . . . . . . . . . . . . . . 125 13 Exercises
Preface This book is based on notes the author created for a one-semester undergraduate course on Algebraic Number Theory, which the author taught at Harvard during Spring 2004 and Spring 2005. This book was mainly inspired by the [SD01, Ch. 1] and Cassels’s article Global Fields in [Cas67]
————————— - Copyright: William Stein, 2005, 2007.
License: Creative Commons Attribution-Share Alike 3.0 License Please send any typos or corrections to [email protected]
Acknowledgement: This book closely builds on Swinnerton-Dyer’s book [SD01] and Cassels’s article [Cas67]. Many of the students of Math 129 at Harvard during Spring 2004 and 2005 made helpful comments: Jennifer Balakrishnan, Peter Behrooz, Jonathan Bloom, David Escott Jayce Getz, Michael Hamburg, Deniz Kural, Danielle li, Andrew Ostergaard, Gregory Price, Grant Schoenebeck, Jennifer Sinnott, Stephen Walker, Daniel Weissman, and Inna Zakharevich in 2004; Mauro Braunstein, Steven Byrnes, William Fithian, Frank Kelly, Alison Miller, Nizameddin Ordulu, Corina Patrascu, Anatoly Preygel, Emily Riehl, Gary Sivek, Steven Sivek, Kaloyan Slavov, Gregory Valiant, and Yan Zhang in 2005. Also the course assistants Matt Bainbridge and Andrei Jorza made many helpful comments. The mathemtical software [S+ 11], [PAR], and [BCP97] were used in writing this book.
This material is based upon work supported by the National Science Foundation under Grant No. 0400386.
In addition to general mathematical maturity, this book assumes you have the following background: • • • • • •
Basics of finite group theory Commutative rings, ideals, quotient rings Some elementary number theory Basic Galois theory of fields Point set topology Basic of topological rings, groups, and measure theory
For example, if you have never worked with finite groups before, you should read another book first. If you haven’t seen much elementary ring theory, there is still hope, but you will have to do some additional reading and exercises. We will briefly review the basics of the Galois theory of number fields. Some of the homework problems involve using a computer, but there are examples which you can build on. We will not assume that you have a programming background or know much about algorithms. Most of the book uses Sage http://sagemath.org, which is free open source mathematical software. The following is an example Sage session: sage: 2 + 2 4 sage: k. = NumberField(x^2 + 1); k Number Field in a with defining polynomial x^2 + 1
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