3510.001, Fall 2003
Introduction to Abstract Algebra
Conley
the University of North Texas. |
INSTRUCTOR: Charles Conley, GAB 419, (940) 565-3326
OFFICE HOURS: MW 12:30-2:00, F 12:30-1:30
CLASS MEETS: MWF 11:00-11:50, GAB 206
EXAMS, HOMEWORK, AND GRADING: There will be two 100 point midterms, on Wednesday, Sept. 24 and Wednesday, Oct. 29, and a comprehensive 180 point final, on Friday, Dec. 12, 10:30-12:30. There will be thirteen homeworks, due at the beginning of the last day of class each week. They will be worth 10 points each, excepting those due the first week and the week of Thanksgiving, which will be worth 5 points each. There will be no make-up exams except for emergencies, and late homework will be worth half-credit.
TEXT AND PREREQUISITES: The text is A first course in Abstract Algebra, 6th edition, by J. Fraleigh. The prerequisite is Math 2510-20, the analysis sequence.
TOPICS: We will cover two main topics, groups and rings. The section of the course on groups will be an introduction to their structure, with an emphasis on examples such as the cyclic and dihedral groups, the groups of symmetries of the Platonic solids, and the symmetric groups. The section on rings will begin with arithmetic in the integers, the unique factorization theorem, and modular arithmetic. Then we will discuss more general rings, emphasizing the particular case of polynomial rings and their quotients, which lead to field theory. Prior experience with analytic proofs will be assumed; here you will be introduced to algebraic proofs.
FINAL EXAM: Friday, Dec.12, 10:30-12:30
HOMEWORK 13, due Friday, December 5
Section 5.3: 1-10, 24-28
Extra Problems:
(A) Find an isomorphism from the additive group Z_{16}to
the multiplicative group Z_{17}^*.
(B) Find all generators of Z_{17}^*
(C) Find 1/6 in Z_{17}
(D) Find all square roots of 2 in Z_{17}
(E) Find all square roots of 6 in Z_{17}
HOMEWORK 12, due Wednesday, November 26 (This homework is
worth only
5 points, not the usual 10)
Section 5.2: 2-4, 8-10, 12, 27
HOMEWORK 11, due Friday, November 21
Section 5.1: 2, 4, 6, 12, 14-16, 19, 28, 32, 38, 40, 49, 52, 55
HOMEWORK 10, due Friday, November 14
Section 3.1: 25, 28, 29, 33-38, 41, 42, 49
Section 3.2: 31, 34*
EXTRA PROBLEMS:
In problems (A)-(D), prove that no surjective
homomorphism exists from
(A) S_3 to Z_3, (B)
A_4 to Z_2, (C) A_4 to any 4 element group, (D) S_4 to any 8
element group.
In problems (E)-(H), argue geometrically that surjective
homomorphisms exist and compute their kernels:
(E) S_3 to Z_2, (F) A_4 to Z_3, (G) S_4 to S_3, (H) S_4
to S_2.
HOMEWORK 9, due Friday, November 7
Section 2.4: 1-5, 9, 11, 15-20
Section 3.1: 4, 5, 16-21, 44, 45, 47
Extra problems: These are postponed
until next week.
EXAM 2: Wednesday, October 29
HOMEWORK 8, due Friday, October 24
Section 0.2: 34-36
Section 2.3: 1-4, 6-12, 16, 26-30, 33, 35, 38, 39
HOMEWORK 7, due Friday, October 17
Section 2.1: 18-20, 23-26, 42-44
Section 2.2: 1-3, 7-18, 24ab, 31
HOMEWORK 6, due Friday, October 10
Section 1.5: 16-20, 34-39, 52-54, 64, 65a
Section 2.1: 1-9, 11-13, 16-17
HOMEWORK 5, due Friday, October 3
Section 1.4: 36, 41, 42, 46, 47, 49, 55
Section 1.5: 7, 9, 11, 12-15, 21, 22, 26-33
EXAM 1: Wednesday, September 24
HOMEWORK 3, due Friday, September 12
Section 1.2: 16, 17, 25, 26, 29, 30, 31
Section 1.3: 1-6, 10
HOMEWORK 2, due Friday, September 5
Section 1.1: 1-7
Section 1.2: 2-7, 10
HOMEWORK 1, due Friday, August 29:
Section 0.1: 29, 30
Section 0.2: 3
Section 0.3: 1, 3, 4, 5, 8b