Lectures: Mondays, Wednesdays, 2:00-3:15PM, in CIWW (Courant Institute) 317.
Office hours: Mondays 3:30-4:30pm; Wednesdays 11:30-12:30am. Office WWH926.
If needed, possibility to set up appointments by email (thomasl@math.nyu.edu)
Recitation sessions: Fridays, 2:00-3:15PM in CIWW 512.
Course description: An introduction to abstract algebra. We will study the fundamental algebraic structures: groups, rings, fields.
Prerequisites: At least one "proof-based" class. The equivalent of discrete mathematics would be helpful. We will give examples and applications, but this is certainly an abstract course.
Textbook(s): We will use the open source textbook "Abstract algebra: Theory and applications." by Thomas W. Judson. You can read it online and download it as a PDF, for free, from the book webpage.
A classical reference that you may consult is Algebra by Michael Artin.
Good lecture notes ("Abstract Algebra", Alexander Paulin, Berkeley University)
Grading: Homework (25%), Midterm 1 (20%), Midterm 2 (20%), Final exam (35%).

• Groups: definitions, subgroups, cyclic groups, permutation groups, group (iso)morphisms.
• Rings: definitions, examples, ideals, integral domains, principal rings.
• Fields: definitions, fields of fractions.

Remark: you may want to check the webpage for other sections of the same class, taught by Liming Pang, here. Syllabus is a bit different from ours.

• Sept 5: Welcome, class logistics. Discussion about structures.
  Homework for 09/10: Read Chapter 1 of textbook
• Sept 10: Groups I (definitions, examples, first properties). Sections 3.1/3.2.
• Homework 1 due Friday, Sept 21.
  Solutions HW1
• Sept 12: Groups II (more properties). Subgroups. Sections 3.2/3.3.
• Sept 17: Cyclic groups. Section 4.1
• Homework 2 due Friday, Sept 28.
  Solutions HW2
• Sept 19: Cyclic groups (end).
• Sept 24: Permutations. Section 5.1
• Homework 3 due Friday, Oct 5.
  Solutions HW3
• Sept 26: Permutations.
• Oct 1: End of permutations. Morphisms, isomorphisms. Sections 9 and 11.1
• Oct 3: Isomorphisms. Subgroup generated by finitely many elements, direct product of groups. Sections 9.2 and 10.1
• Homework 4 due Friday, Oct 12.
  Solutions HW4
• Oct 8: No class.
• Oct 9 [Legislative day - Monday on Tuesday]: Quotient group. Chapter 10.
• Homework 5 due Friday, Oct 19.
  Solutions HW5
• Oct 10: Group actions (Section 14.1), Cayley's theorem (chapter 9).
• Oct 15: Midterm 1.
  Exam and solutions.
• Lecture notes for the “rings” chapter (in progress).
• Oct 17: Rings : definitions, examples. Section 16.1
• Oct 22: Ideals. Section 16.3.
• Oct 24: Principal, prime and maximal ideals. Sections 16.3 and 16.4
• Homework 6 due Friday, Nov 2nd.
  Solutions HW6
• Oct 29: Examples 1: Integers.
• Homework 7 due Friday, Nov 9th.
  Solutions HW7
• Oct 31: Examples 2: rings of functions
• Nov 5: Examples 2: sequences (continued)
• Nov 7: Examples 3: rings of matrices.
• Nov 12: Example 4: Polynomials.
• Nov 14: Midterm 2.
  Sources for practice
  Exam, solutions and some comments.
• Nov 19: Cancelled.
• Nov 21: Thanksgiving.
• Nov 26: Integral domain, divisor of zeroes, invertible elements, fields. Sections 16.2, 16.4.
  Some notes
• Homework 8 due Friday, Dec 7th.
  Solutions HW8
• Nov 28: Connection with quotient rings and prime/maximal ideals. Section 16.2
• Dec 3: Field of fractions Section 18.1.
• Dec 5: Extension fields. Section 21.1.
• Homework 9 due Friday, Dec 14th.
  Solutions HW9
• Dec 10: Algebraicity, algebraic closure. Section 21.1
• Dec 12: Wrap-up about fields.
  How to construct the field of complex numbers?
• Dec 17: Final exam, 2:00pm - 3:50pm.
• Syllabus for final exam
  Review session of Dec 12th
  Final exam.