Math 418, Intro to Abstract Algebra II

Spring 2010

Course Description

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


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.


Your course grade will be based on:

Homework Assignments

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.

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