Algebraic Number Theory II   Syllabus  (Graduate course,   Fall 2019)

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Professor: Yoonjin Lee

Office: Science Complex Building B, 313

Text Book:
Number Fields (Graduate Texts in Mathematics) by D. Marcus, Springer

Reference:
A Classical Introduction to Modern Number Theory by K. Ireland and M. Rosen (2nd ed.), Springer

Course requirements:
Elementary Number Theory,   Abstract Algebra,   Algebraic Number Theory II

Course Description:
Number Theory has been studied for its long and rich history, its wealth of easily accessible and fascinating questions and its intellectual appeal, but even for practical applications to Cryptology and Coding Theory in recent years. Algebraic Number Theory has many attractive and instructive topics. Throughout this course we study comprehensive and advanced concepts of Algebraic Number Theory, assuming students have some background of Elementary Number Theory and Abstract Algebra (Groups, Rings and Fields). The following topics plan to be covered throughout this semester even though some other topics may be added and some topics may be omitted.

Course Outline :
The Gaussian Integers, Integrality
Integrality, Review of some Field Theory
Traces and Norms, Discriminants, Integral Bases
Integral Bases
Ideals, Dedekind Domains
Lattices, Minkowski Theory
The Class Number, Ideal Class Group
Dirichlet's Unit Theorem
Extension of Dedekind Domains
Hilbert's Ramification Theory
Cyclotomic Fields
The p-adic numbers, The p-adic Absolute Value
Valuations, Completions, Hensel's Lemma

Grading Scheme:
Presentation, HW Assignments : 50%
Final Project : 50%

Grades and policy:
*  You will be evaluated throughout the whole semester by means of Homeworks, Presentations, and the final project.