Algebra II: Rings and Fields

Math 123 - Harvard University - Spring 2023
Instructor: Curtis T McMullen

Required Text: Artin, Algebra. Second Edition, Pearson, 2010.

Prerequisites: Math 122 or equivalent. Intended for undergraduate math majors.

Topics. This course will cover the basics of rings and fields. Topics include:
  • Supplement to Algebra I: groups presentations, quadratic forms
  • Rings
  • Integral domains and fields
  • Ideals. Prime and maximal ideals.
  • Algebraic geometry (introduction)
  • Unique factorization domains
  • Integral polynomials. Gauss's lemma.
  • Irreducible polynomials; Eisenstein's criterion
  • Number theory (introduction)
  • Factorization in Z[i]
  • Factorization in imaginary quadratic fields
  • Units in real quadratic fields
  • Modules
  • Diagonalization
  • Fields extensions
  • Finite fields
  • Ruler and compass constructions
  • Galois Theory
  • Solving equations by radicals
  • Quintic polynomials
Readings and Lectures. Assigned material should be read before coming to class.

Lectures may go beyond the reading, and not every topic in the reading will be covered in class. Students are responsible for all topics covered in the readings and lectures.

Technology. Notebooks and tablets may be used for taking notes. The use of laptops and cell phones during class is not permitted. Please do not make sound or video recordings or take pictures in class.

Homework. Weekly homework assignments will be due at the start of class on Tuesdays.

Late homework and homework sent by email will not be accepted.

Collaboration between students on homework is encouraged, but you must write your own solutions, understand them and give credit to your collaborators.
Only materials from this course can be referred to for the homework; e.g. searching for solutions on the web is not permitted.

Problem Sessions . There are four 90 minutes problem sessions run by the course assistants each week. Attending problem sessions is an essential part of the course, especially for learning to write proofs and for establishing collaborations.

Midterm. There will be one in-class midterm, tenatively scheduled for Thursday, 23 March. (Not to conflict with Housing Day, 9 March.)

Final. There will be a take-home final, during the first week after classes end (1-5 May). Collaboration on the final is not permitted. You may refer to your graded homeworks, course notes and the texts for the course, but no other materials. Internet resources CANNOT be consulted during the final, except for the course homepage.

Grading. Grades will be based on regular weekly homework (40%), the midterm (20%) and the final (40%). The two lowest homework grades will be dropped. This course is not offered pass-fail.


Course home page: http://math.harvard.edu/~ctm/math101