Simon Fraser University
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Course Outlines

Spring 2024 - MATH 842 G100

Algebraic Number Theory (4)

Class Number: 3801

Delivery Method: In Person

Overview

  • Course Times + Location:

    Jan 8 – Apr 12, 2024: Wed, 12:30–2:20 p.m.
    Burnaby

    Jan 8 – Apr 12, 2024: Fri, 12:30–2:20 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:

Algebraic number theory comprises the study of algebraic numbers: numbers that satisfy polynomial equations with rational coefficients. The parallels with usual integer arithmetic are striking, as are the notable differences (as, for instance, failure of unique factorization into prime factors). The subject is fundamental to any further study in number theory or algebraic geometry.

In this course we develop the tools to properly understand unique factorization and its failure. We establish fundamental results such as Dirichlet's Unit theorem and the finiteness of the ideal class group. We highlight the applicability of the algebraic tools we develop to both algebraic numbers and to algebraic curves. Depending on time and interests of the participants, we will also look into various applications and more advanced topics.

Grading

  • Biweekly assignments (weighted equally) 30%
  • Presentation 15%
  • Final Examination (take home) 55%

REQUIREMENTS:

Prerequisite: MATH 440/740

Materials

REQUIRED READING:

Milne, J.S.
Algebraic Number Theory
available from: http://www.jmilne.org/math/CourseNotes/ant.html


RECOMMENDED READING:

Alternative and additional reading:  

Quite algebraic and very careful exposition, including lots of detail in proofs:  
Ribenboim, Paulo
Classical theory of algebraic numbers.
Universitext. Springer-Verlag, New York, 2001.
xxiv+681 pp.
ISBN: 0-387-95070-2  

Very compact in its exposition, but still complete. Probably one of the slickest presentations available:

Neukirch, Jürgen
Algebraic number theory.
Grundlehren der Mathematischen Wissenschaften, 322. Springer-Verlag, Berlin, 1999. xviii+571 pp.
ISBN: 3-540-65399-6  

Quite complete in its coverage of theory but also paying attention to the computational side of things:  

Stein, William
Algebraic Number Theory, a Computational Approach
Full text available from: https://wstein.org/books/ant/

REQUIRED READING NOTES:

Your personalized Course Material list, including digital and physical textbooks, are available through the SFU Bookstore website by simply entering your Computing ID at: shop.sfu.ca/course-materials/my-personalized-course-materials.

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:

ACADEMIC INTEGRITY: YOUR WORK, YOUR SUCCESS

SFU’s Academic Integrity website http://www.sfu.ca/students/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