# Math 117: General Course Outline

## Course 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(s)

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 |