Course Description
Abstract algebra: groups, rings, fields.
Prequisite: Math 331 (Linear Algebra).

Texts

• Abstract Algebra, notes by Irena Swanson.
• Algebraic Curves, by William Fulton.

Books on Reserve

• Abstract Algebra, Dummit and Foote
• Contemporary abstract algebra, Joseph Gallian
• Topics in algebra, I. N. Herstein
• Algebra, Serge Lang
• Undergraduate Algebra, Serge Lang
• Algebra, Thomas Hungerford
• Algebra, Michael Artin

• No midterm.
• The final exam will be Monday, May 11, 9–12 noon, P240A. See the review page.

Week 13

• Tuesday.
  • Turn in HW12.
  • Reading: Sections 41 and 42.
  • Links
    • field theory
    • algebraic extension
    • algebraic closure
• Thursday.
  • Reading: class notes from Tuesday and sections 41 and 42.
  • Quiz: In the following, let K be an extension field of k.
    • Let α be an element of K. What does it mean to say that α is algebraic over k?
    • Show that α in K is algebraic over k if and only if k[α]=k(α).
    • What does it mean to say that K is an algebraic extension of k?
    • What does it mean to say that K is algebraically closed?
  • Links
    • Galois theory
    • fundamental theorem of Galois theory
    • Evariste Galois
    • another biography of Galois
    • cubic equation
    • quartic equation
    • quintic equation

• GAP is a system for computational discrete algebra, with particular emphasis on computational group theory.
• CoCoA is a program for computations in commutative algebra. We will use this program during the ring theory part of the course.
• Sage, an open-source alternative to Mathematica or Maple.
  • Tips for using Sage.
  • Group theory in Sage.
  • Math 332 Sage account

Assignments

• Due Tuesday, February 3.
  • HW1. As part of this assignment, you are asked to fill in this following multiplication table. Solutions.
  • Reading assignment: sections 3–7, pp. 4–8.
• Due Tuesday, February 10.
  • HW 2. Solutions.
  • Reading assignment: section 12. Concentrate on understanding the statements of the results (not the proofs). Make up examples to illustrate the results. Perhaps Sage would be useful. The reading assignment for Thursday will include some of the proofs from section 12.
• Due Tuesday, February 17.
  • HW 3. Solutions.
  • Reading assignment: section 14 and the first page of section 15.
• Due Tuesday, February 24.
  • HW 4. Solutions.
  • Reading assignment: Sections 18 and 19.
• Due Tuesday, March 3.
  • HW 5. Solutions.
  • Reading assignment: Sections 21, 22, and 23.
• Due Tuesday, March 10: s.e.s HW. Solutions.
• Due Thursday, March 12: HW 6. Solutions.
• Due Tuesday, March 24: HW 7. Solutions.
• Due Tuesday, March 31: HW 8. Solutions.
• Due Tuesday, April 7: HW 9. Solutions.
• Due Thursday, April 16: HW 10. Solutions.
• Due Tuesday, April 21: HW 11. Solutions.
• Due Tuesday, April 28: HW 12. Solutions.

Class Summary

Week 1

• Tuesday. Definition of a group. First examples.
• Thursday. Groups of orders 2, 3, and 4. Free groups. Groups presented by generators and relations.
  Due on Thursday:
  • Quiz. Sections 1 and 2. Please read these sections carefully, memorize the definitions, and understand the proofs of the results appearing there.
  • Install Sage on your computer.
  Relevant links for Thursday's class.
  • Sage for Cayley graphs
  • Cayley graph wiki
  • free groups wiki

Week 2

• Tuesday.
  • Turn in HW1.
  • Presentations of groups. Subgroups. Conjugate subgroups.
  • Reading: Sections 3–7, pp. 4–8.
  • Links
    • presentations of groups wiki
    • subgroups wiki
• Thursday. Conjugating permutes elements and subgroups. Cyclic groups.
  • Quiz. On sections 9 and 10.
  • Links
    • conjugation
    • cyclic groups
    • Euler totient function

Week 3

• Tuesday.
  • Turn in HW2.
  • Finish talking about cyclic groups. Discuss permutation groups.
  • Reading: Section 12. Concentrate on understanding the statements of the results (not the proofs). Make up examples to illustrate the results. Perhaps Sage would be useful.
  • Links
    • permutation groups
    • parity of a permutation
• Thursday.
  • Quiz. Know the statements of the results in section 12. Read section 13, on group homomorphisms, carefully.
  • Miscellaneous properties of permutations and examples of homomorphisms.

Week 4

• Tuesday.
  • Turn in HW3.
  • Group actions.
  • Reading: section 14 and the first page of section 15.
  • Links
    • group isomorphisms
    • general automorphism groups
    • group actions
• Thursday.
  • Quiz. On sections 15 and 16.
  • Cosets and Lagrange's theorem.
  • Links
    • Lagrange
    • cosets

Week 5

• Tuesday.
  • Turn in HW4.
  • Reading: Sections 18 and 19.
  • Links
    • group actions
    • Burnside's lemma
    • William Burnside
• Thursday.
  • Quiz. Know the definitions and the statements of main results in sections 18, 19, and 20.
  • Links
    • the class equation
    • Chinese remainder theorem

Week 6

• Tuesday.
  • Turn in HW5.
  • Reading: Sections 21, 22, and 23.
  • Links
    • normal subgroups
    • quotient groups
    • reflection groups
    • 3D point groups
    • crystallographic groups
    • finite simple groups
    • sporadic simple groups
    • finite simple groups of order two
• Thursday.
  • Quiz. The quiz covers the reading from sections 21&endash;25, inclusive. Know the statements of the results. Be able to prove the first isomorphism theorem (Thm. 24.1).
  • Links
    • isomorphism theorems
    • exact sequences

Week 7

• Tuesday.
  • Turn in the short exact sequence HW.
  • Reading: Section 25.
  • Links.
    • finitely generated abelian groups
    • Smith normal form
    • Henry Smith
• Thursday.
  • Turn in HW 6.
  • Links
    • semidirect products
    • wreath products
    • semidirect products

Week 8

• Tuesday.
  • Turn in HW7.
  • Reading: sections 27–31, inclusive.
  • Links.
    • rings
• Thursday.
  • Reading: sections 32–34, inclusive.
  • Quiz: Over sections 27 to 34.
  • Links
    • prime ideal
    • maximal ideal
    • An excellent introduction to algebraic geometry: Algebraic Curves, by Bill Fulton
    • algebraic geometry
    • algebraic variety

Week 9

• Tuesday. (Extra notes here.)
  • Turn in HW8.
  • Reading: Read the first PCMI handout on algebraic geometry.
• Thursday. (Extra notes here.)
  • Reading: Second PCMI handout. First three pages of Fulton's book (see link under Texts, above).
  • Quiz: Covers the reading for Tuesday and Thursday this week.

Week 10

• Tuesday. (Extra notes here.)
  • Turn in HW9.
  • Reading: Read the third PCMI handout on algebraic geometry.
• Thursday.
  • Reading: Sections 34.
  • Quiz: Covers the reading for Tuesday and Thursday this week.

Week 11

• Tuesday.
  • Reading: Sections 35 and 36.
  • Links
    • integral domains, irreducibility, etc.
    • PID
    • UFD
• Thursday.
  • Turn in HW10.
  • Quiz. Let R be a commutative ring with 1, and let I be an ideal in R.
    • Prove that I is prime iff R/I is a domain.
    • Prove that I is maximal iff R/I is a field.
    • What does it mean to say that and element r of R is prime? Irreducible?
    • Prove that in a domain, a prime element is irreducible.
  • Links
    • Ferdinand Eisenstein

Week 12

• Tuesday.
  • Turn in HW11.
  • Reading: Gröbner bases.
  • Links
    • Gröbner basis
    • Macaulay
    • Some computer algebra systems for commutative algebra (besides Sage):
      • CoCoA
      • Macaulay 2
      • Singular
• Thursday.
  • Reading: Gröbner basis handout from Tuesday.
  • Quiz: No quiz.

Miscellaneous Links

• Reed College Mathematics Colloquium
• Online Encyclopedia of Integer Sequences
• Old summary of group theory from the last time I taught abstract algebra (1994).
• word game

Week 13

• Tuesday.
  • Turn in HW12.
  • Reading: Sections 41 and 42.
  • Links
    • field theory
    • algebraic extension
    • algebraic closure
• Thursday.
  • Reading: class notes from Tuesday and sections 41 and 42.
  • Quiz: In the following, let K be an extension field of k.
    • Let α be an element of K. What does it mean to say that α is algebraic over k?
    • Show that α in K is algebraic over k if and only if k[α]=k(α).
    • What does it mean to say that K is an algebraic extension of k?
    • What does it mean to say that K is algebraically closed?
  • Links
    • Galois theory
    • fundamental theorem of Galois theory
    • Evariste Galois
    • another biography of Galois
    • cubic equation
    • quartic equation
    • quintic equation