Lectures on Topics in Algebraic Number Theory

Lectures on Topics in Algebraic Number Theory


Sudhir R. Ghorpade

Department of Mathematics
Indian Institute of Technology, Bombay
Powai, Mumbai 400076 India
E-mail: srg@math.iitb.ac.in

Version 1.1, August 18, 2002


These are the notes of a course of ten lectures given at the Christian-Alberchts-Universität Kiel at Kiel, Germany during December 2000. These lectures were aimed at giving a rapid introduction to some basic aspects of Algebraic Number Theory with as few prerequisites as possible. The Table of Contents below gives some idea of the topics covered in these notes. There are two appendices, first containing author's older notes on Galois Theory and the second reproducing a recent article in Bona Mathematica. These notes have a considerable overlap with the ISANT Notes and to a lesser extent, the ICCTA Notes.


1 Field Extensions 6
1.1 Basic Facts 6
1.2 Basic Examples 9
1.3 Norm, Trace and Discriminant 12
2 Ring Extensions 15
2.1 Basic Processes in Ring Theory 15
2.2 Noetherian Rings and Modules 17
2.3 Integral Extensions 19
2.4 Discriminant of a Number Field 21
3 Dedekind Domains and Ramification Theory 26
3.1 Dedekind Domains 27
3.2 Extensions of Primes 32
3.3 Kummer's Theorem 35
3.4 Dedekind's Discriminant Theorem 37
3.5 Ramification in Galois Extensions 38
3.6 Decomposition and Inertia Groups 40
3.7 Quadratic and Cyclotomic Extensions 42
4 Class Number and Lattices 46
4.1 Norm of an ideal 46
4.2 Embeddings and Lattices 48
4.3 Minkowski's Theorem 52
4.4 Finiteness of Class Number and Ramification 53
Bibliography 56
A Appendix: Notes on Galois Theory 57
A.1 Preamble 57
A.2 Field Extensions 58
A.3 Splitting Fields and Normal Extensions 60
A.4 Separable Extensions 62
A.5 Galois Theory 63
A.6 Norms and Traces 67
B Appendix: Discriminants in Algebra and Arithmetic 70
B.1 Discriminant in High School Algebra 70
B.2 Discriminant in College Algebra 74
B.3 Discriminant in Arithmetic 77
References 82

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