Math 332: Abstract
Algebra 
Spring 2009
David Perkinson
L316, ext. 7417
Office
hours
Course Description
Abstract algebra: groups, rings, fields.
Prequisite: Math 331 (Linear Algebra).
Texts
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
Exams
This Week
Week 13
 Tuesday.
 Turn in HW12.
 Reading:
Sections 41 and 42.
 Links
 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
Software
 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 opensource alternative to
Mathematica or Maple.
Assignments
 Due Tuesday, February 3.
 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.
 Due Tuesday, March 3.
 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.
Week 2
 Tuesday.
 Turn in HW1.
 Presentations of groups. Subgroups. Conjugate
subgroups.
 Reading: Sections 3–7, pp. 4–8.
 Links
 Thursday. Conjugating permutes
elements and subgroups. Cyclic groups.
 Quiz. On sections 9 and 10.
 Links
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
 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
 Thursday.
 Quiz. On sections 15 and 16.
 Cosets and Lagrange's theorem.
 Links
Week 5
 Tuesday.
 Turn in HW4.

 Reading: Sections 18 and 19.
 Links
 Thursday.
 Quiz. Know the definitions and the statements
of main results in sections 18, 19, and 20.

 Links
Week 6
 Tuesday.
 Turn in HW5.

 Reading: Sections 21, 22, and 23.
 Links
 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
Week 7
Week 8
 Tuesday.
 Turn in HW7.
 Reading: sections 27–31, inclusive.
 Links.
 Thursday.
 Reading: sections 32–34, inclusive.
 Quiz: Over sections 27 to 34.
 Links
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
 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
Week 12
 Tuesday.
 Thursday.
 Reading: Gröbner basis handout from Tuesday.
 Quiz: No quiz.
Miscellaneous Links
Week 13
 Tuesday.
 Turn in HW12.
 Reading:
Sections 41 and 42.
 Links
 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