Math 361

Number theory
Spring 2023

Current week   |   Moodle   |   Course information
Instructor: David Perkinson (schedule)
Text: Algebraic Number Theory and Fermat's Last Theorem, by Ian Stewart and David Tall.
Office hours: 2:30–3:30 MW, 3–4 TuTh, or by appointment or drop-in.
Week 1: January 23 - 27
Monday: Introduction. Pythagorean triples. (lecture, tex file, slides)
Wednesday: Rings. (lecture, tex file, slides)
Friday: Field extensions. (lecture, tex file, slides)
  • Reading: Sections 1.3 and 2.1 (skip the proof of Theorem 2.2 for now).
Week 2: January 30 - February 3
Monday: Modules and integral extensions. (lecture, tex file, slides)
  • Reading: Sections 1.5 and 2.3. Skip the proofs in 2.3 the first time through.
Wednesday: Criterion for integrality. The algebraic integers form a ring. (lecture, tex file, slides)
  • Reading: Section 2.3. Read the proofs of Lemma 2.8 and Theorem 2.9.
  • We will have a short quiz at the beginning of class covering the following: quiz topics. (The quiz covers material from last week. In general, the quiz will cover the material from all previous weeks, i.e., it will be cumulative.)
Friday: Integers in a quadratic extension. (lecture, tex file, slides)
Week 3: February 6 - 10
Monday: Discriminants I. (lecture, tex file, slides)
  • Reading: Section 2.2.
Wednesday: Discriminants II. (lecture, tex file, slides)
  • Reading: Section 2.2.
  • We will have a short quiz at the beginning of class covering the following: quiz topics.
Friday: Bases for algebraic integers. (lecture, tex file, slides)
Week 4: February 13 - 17
Monday: Norms and traces. (lecture, tex file, slides)
  • Reading: Section 2.5.
Wednesday: Cyclotomic fields I. (lecture, tex file, slides)
  • Reading: Section 3.2.
  • We will have a short quiz at the beginning of class covering the following: quiz topics.
Friday: Cyclotomic fields II. (lecture, tex file)
Week 5: February 20 - 24
Monday: Noetherian modules. (lecture, tex file, slides)
  • Reading: Section 4.3.
Wednesday: Hilbert Basis Theorem. (lecture, tex file, slides)
  • Reading: Lecture notes.
  • We will have a short quiz at the beginning of class covering the following: quiz topics.
Friday: Snow day.
Week 6: February 27 - March 3
Monday: Catch-up day. (slides)
  • Reading: Review lectures from last week.
Wednesday: Unique factorization. (lecture, tex file) slides)
  • Reading: Skim Sections 4.4–4.7.
  • We will have a short quiz at the beginning of class covering the following: quiz topics.
Friday: Operations on ideals. (lecture, tex file) slides)
Week 7: March 6 - 10
Monday: Midterm. The midterm will cover this material.
  • Reading: None
Wednesday: Dedekind domains. (lecture, tex file) slides)
  • Reading: Section 5.2.
Friday: Fractional ideals. (lecture, tex file) slides)
Spring break: March 13 - 17
No classes this week.

Week 8: March 20 - 24
Monday: Catch up. (Finish work from the week before break.) (slides)
  • Reading: Section 5.2
Wednesday: Smith normal form. (lecture, tex file, slides)
  • Reading: Lecture notes.
  • We will have a short quiz at the beginning of class covering the following: quiz topics.
Friday: Structure theorem for finitely generated abelian groups. (lecture, tex file, slides)
Week 9: March 27 - 31
Monday: The norm of an ideal. (lecture, tex file, slides)
  • Reading: Section 5.3.
Wednesday: Generators for ideals in a number ring. (lecture, tex file, slides)
  • Reading: Sections 5.3 and 10.1. Lecture notes.
  • We will have a short quiz at the beginning of class covering the following: quiz topics.
Friday: Group work factoring ideals in number rings (group problems, tex file)
Week 10: April 3 - 7
Monday: Catch up. Finish last Wednesday's work.
  • Reading: Sections 5.3 and 10.1. Lecture notes.
Wednesday: Minkowski's theorem. (lecture, tex file) slides)
  • Reading: Sections 6.1 and 6.2.
  • We will have a short quiz at the beginning of class covering the following: quiz topics.
Friday: Minkowski's theorem. Four squares theorem. (lecture, tex file, slides)
Week 11: April 10 - 14
Monday: Lattices associated with number fields. (lecture, tex file, slides)
  • Reading: Chapter 8 and Theorem 9.4.
Wednesday: The class group I. (lecture, tex file, slides)
  • Reading: Sections 9.1 and 9.3
  • Turn in idea(s) for final project.
Friday: The class group II. (lecture, tex file, slides)
Week 12: April 17 - 21
Monday: Dirichlet's unit theorem. (lecture, tex file, slides)
  • Reading: Appendix B.4.
Wednesday: Project meetings
Friday: Project meetings.

Week 13: April 24 - 28
Monday: Presentations.
Wednesday: Presentations.
Friday: Presentations.
Here is a link to a sample beamer tex file. (It is based on the slides for Monday, Week 11.)
The \(\LaTeX\) document preparation system