Simon Fraser University
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Course Outlines

Spring 2018 - MATH 842 G100

Algebraic Number Theory (4)

Class Number: 3103

Delivery Method: In Person

Overview

  • Course Times + Location:

    Jan 3 – Apr 10, 2018: Tue, Thu, 10:30 a.m.–12:20 p.m.
    Burnaby

Description

CALENDAR DESCRIPTION:

Review of Galois theory, integrality, rings of integers, traces, norms, discriminants, ideals, Dedekind domains, class groups, unit groups, Minkowski theory, ramification, cyclotomic fields, valuations, completions, applications.

COURSE DETAILS:

  1. Introduction and Integrality. Properties of the Gaussian integers. Integral elements and basic properties. Integral closure. Traces, norms, and discriminants. Integral bases.
  2. Ideals. Dedekind domains. Unique prime factorization. Fractional ideals. Ideal and class groups.
  3. Lattices and Minkowski Theory. Definitions. Minkowski’s Lattice Point Theorem. Minkowski space and Minkowski metric
  4. Class numbers.
  5. Dirichlet’s Unit Theorem.
  6. Extensions.
  7. Ramification Theory.
  8. Cyclotomic Fields.
  9. Localization.
  10. Scheme Theorey.
  11. p-adic Numbers and Absolute Value.
  12. Valuations.
  13. Completions and Henselian Fields.
  14. Extensions of Valuations.

Grading

  • Assignments & Participation 30%
  • Final Exam (Take Home) 70%

REQUIREMENTS:

Prerequisite: MATH 440/740

Materials

REQUIRED READING:

Jurgen Neukirch. Algebraic number theory, volume 322 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder.

RECOMMENDED READING:

M. F. Atiyah and I. G. Macdonald. Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.

Serge Lang. Algebraic number theory, volume 110 of Graduate Texts in Mathematics. SpringerVerlag, New York, second edition, 1994.

Serge Lang. Algebra, volume 211 of Graduate Texts in Mathematics. Springer-Verlag, New York, third edition, 2002.

Dino Lorenzini. An invitation to arithmetic geometry, volume 9 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1996.

Graduate Studies Notes:

Important dates and deadlines for graduate students are found here: http://www.sfu.ca/dean-gradstudies/current/important_dates/guidelines.html. The deadline to drop a course with a 100% refund is the end of week 2. The deadline to drop with no notation on your transcript is the end of week 3.

Registrar Notes:

SFU’s Academic Integrity web site http://students.sfu.ca/academicintegrity.html is filled with information on what is meant by academic dishonesty, where you can find resources to help with your studies and the consequences of cheating.  Check out the site for more information and videos that help explain the issues in plain English.

Each student is responsible for his or her conduct as it affects the University community.  Academic dishonesty, in whatever form, is ultimately destructive of the values of the University. Furthermore, it is unfair and discouraging to the majority of students who pursue their studies honestly. Scholarly integrity is required of all members of the University. http://www.sfu.ca/policies/gazette/student/s10-01.html

ACADEMIC INTEGRITY: YOUR WORK, YOUR SUCCESS