(This post is part of the algebra notes series.)

This post begins a series of notes on Abstract algebra. These notes are based off the book Contemporary Abstract Algebra 7th ed. I'll attempt to make these notes as readable, usable, and study-able as possible.

Well Ordering Principle

Every nonempty set of positive integers contains a smallest member.

Thm 0.1: Division Algorithm

Let a and b be integers with b > 0. Then there exist unique integers q and r with the property that a=bq+r, where 0<=r<b.

Thm 0.2: GCD as a linear combination

For any nonzero integers a and b, there exist integers s and t such that gcd(a,b)=as+bt. Also, gcd(a,b) is the smallest positive integer of the form as+bt.

Corollary

If a and b are relatively prime, then there exist integers s and t such that as+bt=1.

Euclid's Lemma

If p is a prime that divides ab, then p divides a or p divides b

Thm 0.3: Fundamental Theorem of Arithmetic

Every integer greater than 1 is a prime or a product of primes. This product is unique, except for the order the factors appear.

Modular Arithmetic

Equivalence Relations

A relation(a set of ordered pairs) R on a set S is an equivalence relation if

1. aRa for all a in S (reflexive)
2. aRb implies bRa (symmetric)
3. aRb and bRc implies aRc (transitive)

Partition

A partition of a set S is a collection of nonempty, disjoint subsets of S whose union is S

Functions

Thm 0.7: Properties of Functions

(This post is part of the algebra notes series.)