Dr. Jeanne LaDuke

Office Hours: MWF 10:50-11:50 and by appointment in Byrne 556
Telephone: (W) (773) 325-1342
(H) (773) 327-9313

E-mail: jladuke@condor.depaul.edu
Homepage: http://www.depaul.edu/~jladuke/

Text: Beachy, John A. and William D. Blair, Abstract Algebra, Second Edition, Waveland Press, 1996.

Prerequisites: MAT 612 and 660 or consent of program director.

Class meetings: Saturday, March 25, 9-5; Sunday, April 9, 9-5; Sunday, April 30, 9-5; Saturday, May 20, 9-5; Sunday, June 4, 1-5.

1. Review of elementary number theory (1.1, 1.2, A.4)
Key ideas: proof by mathematical induction, divisibility, greatest common divisor, Euclidean algorithm, prime numbers, relative primeness, unique factorization

2. Rational numbers and real numbers
Key ideas: least upper bound and intermediate value theorem

3. Complex numbers (A.5)
Key ideas: DeMoivre’s theorem

4. Congruences and the integers modulo m (1.3 (23-26); 1.4 (32-37))
Key ideas: Z_m, characteristic p vs. characteristic 0, polynomials over Z_p, homomorphisms

5. Polynomials over the integers and rationals (4.1, 4.2, 4.3)
Key ideas: rational root test, reducibility and irreducibility of polynomials, reduction modulo p, Eisenstein’s test for irreducibility, Gauss’s lemma, multiple roots and applications of the "derivative"

6. Polynomials over fields (4.2)
Key ideas: similarities between polynomials and integers with irreducible polynomials playing the role analogous to prime numbers, greatest common divisor, Euclidean algorithm, unique factorization, roots of polynomials, extension fields.

7. Introduction to linear algebra and vector spaces (A.7)
Key ideas: Basis, dimension, solutions of systems of linear equations

8. Fields (6.1)
Key ideas: construct and obtain experience with many examples

9. Return to linear algebra (4.4, 6.2)
Key ideas: extend earlier concepts to vector spaces over arbitrary fields, applications to field extensions, applications to finite fields

10. Ruler and compass constructions (6.3)
Key ideas: impossibility of the trisection of an angle

Optional topic: Fundamental theorem of algebra

Evaluation: The midterm exam will be on April 30 and will be based on material covered the first two days of class. A take-home final will be distributed in class on May 20 and will be due by June 4. You may work with other people on the final, and you may use outside references (properly noted), but do not write anything you do not understand. Both exams will be based on problem sets that are to be done but not handed in. Each exam will count for 50% of your grade. Informal review sessions will be held before the midterm.