# Math 117: General Course Outline

## Catalog Description

117. Algebra for Applications. (4) Lecture, three hours; discussion, one hour. Requisite: course 115A. Not open for credit to students with credit for course 110A. Integers, congruences; fields, applications of finite fields; polynomials; permutations, introduction to groups.

## Course Information:

The following schedule is based on 26 lectures. The remaining three classroom meetings are for midterm exams and a review.

Math 117 is the "fast" course in abstract algebra, which focuses on topics that are of interest for applications. The topics covered include error correcting codes, fast polynomial multiplication, and the fast Fourier transform. The fast Fourier transform is absolutely critical for the efficient implementation of computer algorithms for signal processing and other engineering applications.

One section of Math 117 is offered each term. In the past several years the enrollments in the course have averaged about 35 students each term.

## Textbook

L. Childs, A Concrete Introduction to Higher Algebra, 3rd Ed., Springer-Verlag.

Note: The book contains a wealth of interesting topics (e.g. Sturm's theorem, group theory), which can be substituted for material in the last five lectures at the instructor's discretion.

Outline update: D. Gieseker, 1/97

## Schedule of Lectures

Lecture Section Topics

1-2

Ch 2 A--D

Induction and binomial theorem

3

Ch 3 A

Division theorem, bases

4-5

Ch 3 B--D; Ch 4 A, B

Euclidean algorithm, Bezout's identity, unique factorization

6-9

Ch 5; Ch 6

Congruences, congruence classes, and error-correcting codes

10-11

Ch 7

Rings and fields

12-13

Ch 9 A--D

Theorems of Euler and Fermat

14

Ch 10 B

RSA codes

15-16

Ch 12 A, B

Chinese remainder theorem

17

Ch 12 C

Application of Chinese remainder theorem to RSA cryptography

18-20

Ch 13, ch 14

Polynomials, unique factorization

21

Ch 15 D, F, C

Complex numbers, fundamental theorem of algebra

22-23

Ch 17 A, B

Congruences modulo a polynomial and Chinese remainder theorem

24-26

Ch 18

Fast polynomial multiplication, fast Fourier transform