Algebraic Number Theory --- Fall 2021

Florida Atlantic University • Fall 2021 • Markus Schmidmeier

Algebraic Number Theory

Fall 2021

Hi, welcome to my course Algebraic Number Theory (MAS 6215 (20580) or MAT 4930 (21663), 3 credits). We meet Wednesdays and Fridays 12:30 - 1:50 p.m. in PS 113.

The theory of numbers is often called the queen of mathematics. The simplicity of the subject matter (the ordinary integers and their generalizations), the elegance and diversity of its methods, its numerous unsolved problems strongly attract mathematicians of all classes, whether they are beginners, undergraduate students, graduate students, professional number theorists or specialists in other areas in mathematics.
In this course, we will meet some algebraic points of view towards number theory, aside from a result by Minkowski concerning lattices in Rⁿ, this establishes the context in which number theoretic problems have their most natural formulation: For example, the problem of finding all integer solution to the Pell-Fermat equation x²-dy²=±1 (d a square-free integer) involves the study of the field Q[√d]... For the ``great'' Fermat equation, xⁿ+yⁿ=zⁿ the field of n-th roots of unity plays an analogous role... In order to represent an integer as a sum of two squares, it is advantageous to work in the ring of Gaussian integers Q[i]...

Prerequisite

An introductory algebra course covering field extensions.

Textbook and Topics

We will use two text books:
Pierre Samuel, Algebraic Theory of Numbers, Dover, ISBN: 978-0-486-46666-8
Robert B. Ash, Algebraic Number Theory, Dover, ISBN: 978-0-486-47754-1
We are going to cover the following topics:

Number fields and their integers (Ash Chapters 1-2, 4 weeks)
Dedekind rings and lifting prime ideals (Ash Chapters 3-4, 5 weeks)
Lattices and Dirichlet's Unit Theorem (Ash Chapters 5-6, 5 weeks)

Objectives

Use concepts from algebra to prove some classical results about numbers;
Utilize field extensions to study number theoretic problems;
Compute integers, prime ideals and units in Dedekind rings.

Credit
Homework: I will assign homework problems every week. Please bookmark the link:
Homework Problems.
Quizzes: We will have a quiz every week on Friday, starting from the second week. The quiz is based on the homework assignments. The best ten quizzes count for 40 % of the grade.
Presentation: Two presentations during class about problems from the book by Samuel will count for 20 % of the grade. Unless there are volunteers, I will assign presentations for the next class meeting at random among all registered students.
Final Exam: The final exam is comprehensive and will count for 40 % of your grade. It is scheduled for Friday, December 10, 10:30 a.m. - 1:00 p.m. in our classroom. Please bring a picture id (Owl card or drivers licence)!

Further Information
For the Disability Policy, the Make-Up Policy, the Code of Academic Integrity, Religious Accommodation, my Grading Scale, the FAU Attendance Policy Statement, Financial Assistance Opportunities and the FAU Covid Statement please visit Infos for all my courses.

Contact Me

Office hours: Mw 2:00 - 3:20 p.m. in SE 272.

Phone: 561-297-0275 (office)

E-mail: markus@math.fau.edu.

Web Site: math.fau.edu/markus

Last modified: by Markus Schmidmeier