Algebraic Number Theory

Instructor: Guanghua JI

Classroom and Time: 203, Old math buliding; 1-3, THU

Description and Prerequisites: This is a standard graduate course in algebraic number theory. We will cover three fundamental theorems(Unique factorization, the finitenesss of the ideal class group and Dirichlet's unit theorem) of ideal theory, Hilbert's theory of Galois extension; valuation theory, local fields, Adele, Idele; Tate's thesis and Artin L-functions etc. The only prerequisites listed for this course are primary arithmetic and undergraduate abstract algebra.

Grading: The course grade will be based homework(50%), final exam(40%) and final project(10%).

Office Hours: Open, Come any time!

References:

(1) J.W.S. Cassels and A. Frohlich, eds., Algebraic Number Theory, LMS, 2010.

(2) L.J. Goldstein, Analytic Number Theory, Prentice-Hall, Inc. Englewood Cliffs, New Jersey, 1971.

(3) S. Lang, Algebraic Number Theory, Springer-Verlag, GTM110.

(4) J.S. Milne, Algebraic Number Theory, available at http://www.jmilne.org/math/

(5) H.P.F. Swinnerton-Dyer, A Brief Guide to Algebraic Number Theory, Cambrige University Press, 2001.

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Contents

Preface ii

1 Ideal Theory

1.1 The Ring of Integers . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Ideals and Factorization . . . . . . . . . . . . . . . . . . . . 9

1.3 Ideal Class Group and Units . . . . . . . . . . . . . . . . . . 19

1.4 Extensions of Fields . . . . . . . . . . . . . . . . . . . . . . . 28

1.5 Global Hilbert Theory . . . . . . . . . . . . . . . . . . . . . 36

2 Valuation Theory

2.1 Valuations and Completions . . . . . . . . . . . . . . . . . . 47

2.2 Local Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.3 Extensions of Valuations . . . . . . . . . . . . . . . . . . . . 67

2.4 Local Hilbert theory . . . . . . . . . . . . . . . . . . . . . . 77

2.5 Local Ramication Theory . . . . . . . . . . . . . . . . . . . 81

3 Adele, Idele and Harmonic Analysis

3.1 Adeles and Ideles . . . . . . . . . . . . . . . . . . . . . . . . 89

3.2 Characters on local and global fields . . . . . . . . . . . . . 97

3.3 Haar measures and Fourier transform . . . . . . . . . . . . . 106

3.4 The Bruhat-Schwartz space and PSF . . . . . . . . . . . . . 112

4 Arithmetic L-functions

4.1 Hecke L-functions . . . . . . . . . . . . . . . . . . . . . . . . 117

4.2 Tate's thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.3 Applications of Hecke L-functions . . . . . . . . . . . . . . . 128

4.4 Artin L-functions . . . . . . . . . . . . . . . . . . . . . . . . 136

Appendices

A.1 Algebraic Preliminaries . . . . . . . . . . . . . . . . . . . . . 142

B.2 Complex Functions Theory . . . . . . . . . . . . . . . . . . . . 154

C.3 Locally Compact Groups . . . . . . . . . . . . . . . . . . . 160

Bibliography

Index

ALL ARE WELCOME