MTH 310 Lecture Notes Based on Hungerford, Abstract Algebra

Logic......Page 5
Sets......Page 10
Relations and Functions......Page 15
The Natural Numbers and Induction......Page 20
Equivalence Relations......Page 24
Definitions and Examples......Page 29
Elementary Properties of Rings......Page 33
The General Associative, Commutative and Distributive Laws in Rings......Page 37
Divisibility and Congruence in Rings......Page 40
Congruence in the ring of integers......Page 47
Modular Arithmetic in Commutative Rings......Page 51
Subrings......Page 58
Units in Rings......Page 61
The Euclidean Algorithm for Integers......Page 66
Integral Primes......Page 70
Isomorphism and Homomorphism......Page 72
Associates in commutative rings......Page 83
Addition and Multiplication......Page 87
Divisibility in F[x]......Page 97
Irreducible Polynomials......Page 106
Polynomial function......Page 111
The Congruence Relation......Page 121
Congruence Class Arithmetic......Page 124
Fp[] when p is irreducible......Page 129
Ideals......Page 133
Quotient Rings......Page 137
Rules of Logic......Page 143
The inverse of a function......Page 149
Partitions......Page 152
Definition of the real numbers......Page 155
Properties of the natural numbers......Page 157
The General Associative Law......Page 159
The general commutative law......Page 160
The General Distributive Law......Page 162
Verifying Ring Axioms......Page 165
Matrix rings......Page 167
Polynomial Rings......Page 170
Cardinalities of Finite Sets......Page 175