# Lecture 1: The Integers, Modular Arithmetic, and the Euclidean Algorithm

### Date: Jan. 5

### Background Material and Further Resources

- Asymptotic computing time analysis

### Reading

- Material in Chap. 2 of the text on rings and the Euclidean algorithm
- Handout on the computing time of the Euclidean algorithm

### Topics

- Integers Z
- Commutative Ring
- Integral Domain
- Unique Factorization Domain
- Division with remainder
- Euclidean Domain

- Modular arithmetic Z_n
- Equivalence modulo n
- Equivalence classes and arithmetic with equivalence classes
- Z_p is a field when n is prime
- Z_n has zero divisors when n is not prime

- Rings and Fields (division and units)
- Integral domains and zero divisors (A finite integral domain is a
field)
- Ideals and quotient rings
- (n), ideal generated by n (i.e. multiples of n) is an ideal in Z
- Z_n is the quotient ring Z/(n)

- Factorization and Unique Factorization Domains
- Greatest Common Divisors and the Euclidean algorithm
- Extended Euclidean algorithm
- Computing inverses in Z_n using the extended Euclidean algorithm
- Computing time analysis
- dominance, strict dominance and codominance
- Computer representation of integers and the Length of an integer
- Computing time functions
- Computing time of basic integer arithmetic operations
- Computing time of the Euclidean algorithm

### Assignment

The assignment description contains a summary of the material
introduced in this lecture.
Created: Jan. 5, 2000 (revised Jan. 12) by
jjohnson@mcs.drexel.edu