Algebraic Number Theory

Instructor:Guanghua JI

Classroom and Time:102, Old math buliding; 9-11, 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(unique factorization, Euler function, quadratic reciprocity) and undergraduate abstract algebra(finitely generated abel group, domian ring, field theory, finite field, Galois theory). The main reference book is the Dyer's book "a brief guide to algebraic number theory".

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, Thompson Publishing Co., 1967; 2nd, 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) J. Neukirch, Algebraic Number Theory, GMW, Vol. 322, 1999.

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

****************************************************************************************************************************

Course Syllabus:

I, Ideal Theory

2/15,1. The ring of integers(1) Basic concepts (2) Norms, Traces and Discriminant.Homework

2/22, (3) Noetherian ring.2. Ideals and factorization(1) Dedekind domain, (2) Fractional ideal.Homewoork

2/29, (3) CRT, (4) Norm of ideals.3. The class number and Dirichlet unit theorem (1) Lattices and lattice point theorem.

3/7, (2) Finiteness of the class group, (3) Dirichlet unit theorem, (4) Compute the class group and units.Homework

3/15, 4. Extensions of number fields(1) Prime ideals decomopositions, (2) Relative extensions.Homework

3/22, 5. Global Hilbert theory(1) Decomposition and inertia groups, (2) The Frobenius automorphism.Homework

II, Valuation Theory

3/29,1. Valuations(1) Basic concepts, (2)Valuations on number fields, (3), Product formula, (4), Completions.Homework

4/5, 2. The arithmtic of local fields(1) The Structures of local fields, (3), Hensel's lemma(3) Weak approximation Theorem

4/12,3. Extensions of valuations(1)Normed spaces and tensor product, (2), Extensions of valuations.Homework

4/19, (3), Unramified and ramified extension, (4), Local Hilbert theory.,Homework

4/26, 4. Ramification theory(1) The different, (2) The discrinamint,Homework

5/3, (3) Ramification theory.Homework

III, Adeles, Ideles and Harmonic Analysis

5/10,1. Adele ringand Idele groups(1) Restricted direct product (2) The adele ring (3) The idele groupHomework

5/17,2. Characters on the local and global fields (1) Duality theory (2) Additive characters (3), Multiplicative characters.Homework

5/24,4. Harmonic Analysis on Adele groups(1), Harr measures (2) Fourier transforms (3) The Schwartz-Bruhat space (4) Possion summation formula.Homework

VI, Arithmetic L-funtions

5/31, 1.Tate's thesis(1) Local theory, (2) Global theory,Homework

6/7,2.Dedekind zeta functions and Hecke L-functions,Homework

6/14,3. Artin L-functions,Homework

Other Online Resources

Algebraic Number Theory Final Exam

Algebraic Number Theory Final Project

ALL ARE WELCOME!