# Math 103B Winter 2006

### Instructors

#### Professor:

Name Office E-mail Phone Office Hours Lecture Time Lecture Place
Prof. Daniel Rogalski AP&M 5131 drogalsk@math.ucsd.edu 534-4421 W 4-5 MWF 2-2:50pm WLH 2113

#### Teaching Assistant:

Name Office E-mail Office Hours Section Times Section Place
John Farina AP&M 5018 jfarina@math.ucsd.edu W 3-4 W 7-7:50pm WLH 2110

### General Course Information

Textbook: Contemporary Abstract Algebra, 6th Edition, by Joseph Gallian.

The main topic of this course is the theory of rings and fields. We plan to cover most of chapters 12-23 and chapter 31 of Gallian.

There will be 2 in-class midterms on Wed. 2/1/06 and Wed. 3/1/06, and a final exam on Mon 3/20/06 from 3-6pm. No makeup exams will be given.

Homework assignments will be due weekly on Fridays in class.

The final grade will be determined as follows: Homework 25%, Midterms 25%, Final Exam 50%.

More detailed descriptions, including a tentative syllabus, may be found in the first day course handout here: Course syllabus
Check below for more up-to-date information about the schedule of homework and lectures.

### Schedule of Lectures:

1/9/06 Chap 12: Definition of ring, examples.
1/11/06 Chap 12,13: Basic properties of rings. Subrings. (Integral) domains.
1/13/06 Chap 13: Fields. Characteristic.
1/16/06 NO CLASS
1/18/06 Chap 14: Ideals, factor rings.
1/20/06 Chap 14: Examples of factor rings. Prime and maximal ideals.
1/23/06 Chap 15: Homomorphisms. 1st homomorphism theorem.
1/25/06 Chap 15: The characteristic homomorphism Z to R for a ring R. Ideals = kernels.
1/27/06 Chap 15: Field of quotients of a commutative domain.
1/30/06 Chap 16: Introduction to polynomial rings. + Exam Review.
2/1/06 Exam I
2/3/06 Chap 16: Division algorithm for polynomials.
2/6/06 Chap 16: F[x] is a PID. Evaluation. R[x]/(x^2 + 1) is isomorphic to C.
2/8/06 Chap 16,17: Remainder and Factor theorems. A polynomial of degree n has at most n roots. Irreducible polynomials.
2/10/06 Chap 17: Irreducibility tests.
2/13/06 Chap 17: Proof of mod p test. Principal ideals (f) are maximal if and only if f is irreducible. Construction of finite fields.
2/15/06 Chap 18: Irreducibility and prime elements in rings, UFDs.
2/17/06 Chap 18: Rings of the form Z[sqrt(d)].
2/20/06 NO CLASS
2/22/06 Chap 18: A PID is a UFD. Euclidean domains.
2/24/06 Chap 18: Z[i] is a Euclidean domain. Z[x] is not a PID.
2/27/06 Exam Review
3/1/06 Exam II
3/3/06 Chap 19: Vector Spaces
3/6/06 Chap 20-21: Field extensions. Splitting Fields over Q.
3/8/06 Chap 20-21: Big Theorem on Field extensions F(alpha).
3/10/06 Chap 31: Error correcting codes part I
3/13/06 Chap 31: Error correcting codes part II
3/15/06 Insolvability of the Quintic
3/17/06 Review day
3/20/06 FINAL EXAM

### Homework Assignments and Exam review sheets:

Homework #1, due 1/13/06

Homework #2, due 1/20/06

Homework #3, due 1/27/06

Exam 1 review sheet
Exam 1 solutions (includes exam)

Homework #4, due 2/10/06

Homework #5, due 2/17/06

Homework #6, due 2/24/06

Exam 2 review sheet
Exam 2 solutions (includes exam)

Homework #7, due 3/10/06

Homework #8, due 3/17/06

Final Exam review sheet