Topics in Algebraic Number Theory

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During December 2000, I gave a course of ten lectures on Algebraic Number Theory at the University of Kiel in Germany. These lectures were aimed at giving a rapid introduction to some basic aspects of Algebraic Number Theory with as few prerequisites as possible. I had also hoped to cover some parts of Algebraic Geometry based on the idea, which goes back to Dedekind, that algebraic number fields and algebraic curves are analogous objects. But in the end, I had no time to discuss any Algebraic Geometry. However, I tried to be thorough in regard to the material discussed and most of the proofs were either explained fully or at least sketched during the lectures. These lecture notes are a belated fulfillment of the promise made to the participants of my course and the Kieler Graduiertenkolleg. I hope that they will still be of some use to the participants of my course and other students alike. The first chapter is a brisk review of a number of basic notions and results which are usually covered in the courses on Field Theory or Galois Theory. A somewhat detailed discussion of the notion of norm, trace and discriminant is included here. The second chapter begins with a discussion of basic constructions concerning rings, and goes on to discuss rudiments of noetherian rings and integral extensions. Although both these chapters seem to belong to Algebra, they are mostly written with a view towards Number Theory. Chapters 3 and 4 discuss topics such as Dedekind domains, ramification of primes, class group and class number, which belong more properly to Algebraic Number Theory. Some motivation and historical remarks can be found at the beginning of Chapter 3. Several exercises are scattered throughout these notes. However, I have tried to avoid the temptation of relegating as exercises some messy steps in the proofs of the main theorems. A more extensive collection of exercises is available in the books cited in the bibliography, especially [4] and [13]. In preparing these notes, I have borrowed heavily from my notes on Field Theory and Ramification Theory for the Instructional School on Algebraic Number Theory (ISANT) held at Bombay University in December 1994 and to a lesser extent, from my notes on Commutative Algebra for the Instructional Conference on Combinatorial Topology and Algebra (ICCTA) held at IIT Bombay in December 1993. Nevertheless, these notes are neither a subset nor a superset of the ISANT Notes or the ICCTA notes. In order to make these notes self-contained, I have inserted two appendices in the end. The first appendix contains my Notes on Galois Theory, which have been in private circulation at least since October 1994. The second appendix reproduces my recent article in Bona Mathematica which gives a leisurely account of discriminants. There is a slight repetition of some of the material in earlier chapters but this article may be useful for a student who might like to see some connection between the discriminant in the context of field extensions and the classical discriminant such as that of a quadratic. It is a pleasure to record my gratitude to the participants of my course, especially, Andreas Baltz, Hauke Klein and Prof. Maxim Skriganov for their interest, and to the Kiel graduate school “Efficient Algorithms and Multiscale Methods” of the German Research Foundation (“Deutsche Forschungsgemeinschaft”) for its support. I am particularly grateful to Prof. Dr. Anand Srivastav for his keen interest and encouragement. Comments or suggestions concerning these notes are most welcome and may be communicated to me by e-mail. Corrections or future revisions to these notes will be posted on my web page at http://www.math.iitb.ac.in/∼srg/Lecnotes.html and the other notes mentioned in the above paragraph will also be available here.