Math 418, Intro to Abstract Algebra II

Spring 2010

• Time and Place: MWF at 9am in 345 Altgeld Hall.
• Section: B13 CRN: 38013
• Instructor: Nathan Dunfield
  • E-mail:
  • Office: 378 Altgeld Office Phone: (217) 244-3892
  • Office Hours: Mondays from 10-11, Tuesday from 3-5, and by appointment. For an appointment, just talk to me after class or send email.
• Web page: http://dunfield.info/418/
• Homework assignments
• Lecture notes

Course Description

The course content will be tailored for the students taking it. However, the standard syllabus includes:

• Rings: Polynomial rings, fields of fractions, and other examples. Euclidean domains, principal ideal domains, and unique factorization domains.
• Fields: Field extensions and Galois Theory. Solvability of equations by radicals. Ruler and compass constructions.
• Modules: Finitely generated modules over a principal ideal domain. Applications to finitely generated groups over principal ideal domains.
• Other topics: These may include:
  • Representation theory of finite groups.
  • Algebraic geometry.
  • Error-correcting codes.

Prerequisites:

The needed background for this course is Math 417, Intro to Abstract Algebra. Math 427 is also fine, though there is some overlap between that course and this one.

Required text: Dummit and Foote, Abstract Algebra, 3rd Edition, 944 pages, Wiley 2003.

You can get it from Amazon for $65, and bit less at some smaller online shops. The bookstore is considerably more expensive ($111 new, $100 used).

Supplementary texts: For the final part of the course covering algebraic geometry, one good reference is beyond Chapter 15 of Dummit and Foote is:

Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms, Springer Undergraduate Texts in Mathematics.

You can get it in PDF format via the Library's e-book collection here. Another nice book is:

Reid, Undergraduate Algebraic Geometry, London Math. Soc. Student Texts #12.

Grading

• Weekly homework assignments: (20%) These will typically be due in class on Wednesday. Late homework will not be accepted; however, your lowest two homework grades will be dropped, so you are effectively allowed two infinitely late assignments. Collaboration on homework is permitted, nay encouraged. However, you must write up your solutions individually and understand them completely.
• Two takehome midterms: (12.5% each) These are glorified HW assignments that you are to work on individually. They will replace the usual HW for two weeks of the term.
• In class midterm: (20%) This one-hour exam will be held in our usual classroom, on Friday, March 5.
• Final exam: (35%) This will be Friday, May 7 from 1:30-4:30.

Homework Assignments

• HW 1. Due Wed, Jan 27.
• HW 2. Due Wed, Feb 3.
• HW 3. Due Wed, Feb 10.
• Takehome #1. Due Wed, Feb 17.
• HW 4. Due Wed, Feb 24.
• HW 5. Due Wed, March 3.
• In class midterm. Friday, March 5.
• HW 6. Due Friday, March 19.
• HW 7. Due Wed, April 7.
• HW 8. Due Wed, April 14.
• Takehome #2. Due Wed, April 21.
• HW 9. Due Wed, April 28.
• HW 10. Due Wed, May 5.

Lecture notes

Here are scans of my lecture notes, in PDF format.

1. Jan 20: Introduction.
2. Jan 22: Euclidean Domains.
3. Jan 25: Principal Ideal Domains.
4. Jan 27: PIDs are UFDs.
5. Jan 29: Which polynomial rings are UFDs?
6. Feb 1: R[x] is a UFD if R is. Irreducibility criteria.
7. Feb 3: Field extensions I.
8. Feb 5: Field extensions II.
9. Feb 8: Algebraic numbers and extensions.
10. Feb 10: More on algebraic extensions.
11. Feb 12: Field multiplication as linear transformations.
12. Feb 15: Limitations of straightedge and compass.
13. Feb 17: Constructable numbers; Splitting fields.
14. Feb 19: Splitting fields II.
15. Feb 22: Algebraically closed fields; polynomials with distinct roots.
    Supplement: A topological proof of the Fundamental Theorem of Algebra.
16. Feb 24: Criterion of separability; finite fields.
17. Feb 26: A proof of the Fundamental Theorem of Algebra.
18. Mar 1: Rest of the proof of the FTA; Degrees of cyclotomic fields.
19. Mar 3: Cyclotomic polynomials and applications.
20. Mar 8: Introduction to Galois Theory.
21. Mar 10: Galois groups of splitting fields.
22. Mar 12: Primitive extensions and minimal polynomials.
23. Mar 15: Finite fields and degrees of fixed fields.
24. Mar 17: The Fundamental Theorem of Galois Theory.
25. Mar 19: The Fundamental Theorem of Galois Theory II.
26. Mar 29: Possible Galois groups and the discriminant.
27. Mar 31: Galois groups of polynomials.
28. Apr 2: Galois groups of polynomials II.
29. Apr 5: Solvable groups.
30. Apr 7: Extensions with solvable Galois groups.
31. Apr 9: Introduction to Algebraic Geometry.
32. Apr 12: Radical ideals and the Nullstellensatz.
33. Apr 14: Decomposition into irreducibles.
34. Apr 16: Projective space.
35. Apr 19: Projective space II.
36. Apr 21: Elliptic curves.
37. Apr 26: Topology of curves and function fields of varieties.
38. Apr 28: Rational functions and field extensions.
39. Apr 30: Rational functions and field extensions II.
40. May 2: Cayley graphs and branched covers.
41. May 5: The end.