Integers, polynomials, and rings: a course in algebra/ Ronald S. Irving

1 Introduction: The McNugget Problem 1
Part I Integers
2 Induction and the Division Theorem 9
2.1 The Method of Induction 9
2.2 The Tower of Hanoi 15
2.3 The Division Theorem 17
3 The Euclidean Algorithm 23
3.1 Greatest Common Divisors 23
3.2 The Euclidean Algorithm 27
3.3 Bezout's Theorem 31
3.4 An Application of Bezout's Theorem 34
3.5 Diophantine Equations 36
4 Congruences 41
4.1 Congruences 41
4.2 Solving Congruences 46
4.3 Congruence Classes and McNuggets 50
5 Prime Numbers 57
5.1 Prime Numbers and Generalized Induction 57
5.2 Uniqueness of Prime Factorizations 61
5.3 Greatest Common Divisors Revisited 63
6.4 Modular Arithmetic Rings gg
6.5 Congruence Rings gj
7 Euler's Theorem g5
7.1 Units g5
7.2 Roots of Unity gg
7.3 The Theorems of Fermat and Euler 101
7.4 Tlie Euler 0-Function 105
7.5 RSA Encryption HO
8 Binomial Coefficients
8.1 Pascal's Triangle
8.2 The Binomial Theorem 120
Part II Polynomials
9 Polynomials and Roots 127
9.1 Polynomial Equations 127
9.2 Rings of Polynomials 128
9.3 Factoring a Polynomial 130
9.4 The Roots of a Polynomial 133
9.5 Minimal Polynomials 136
10 Polynomials with Real Coefficients 141
10.1
Quadratic Polynomials 141
10.2
Cubic Polynomials 146
10.3 The Discriminant of a Cubic Polynomial 153
10.4 Quartic Polynomials 159
10.5 A Closer Look at Quartic Polynomials 164
10.6 The Discriminant of a Quartic Polynomial 167
10.7 The Fundamental Theorem of Algebra 171
11 Polynomials with Rational Coefficients 177
11.1 Polynomials over Q 177
11.2 Gauss's Lemma 181
11.3 Eisenstein's Criterion 184
11.4 Polynomials with Coefficients in Fp 187
Contents xv
12 Polynomial Rings 193
12.1 Unique Factorization for Integers Revisited 193
12.2 The Finchdean Algorithm 196
12.3 Bezout's Theorem 198
12.4 Unique Factorization for Polynomials 199
13 Quadratic Polynomials 201
13.1 Square Roots 201
13.2
The Quadratic Formula 204
13.3 Square Roots in Finite Fields 209
13.4 Quadratic Field Constructions 214
14 Polynomial Congruence Rings 221
14.1 A Construction of New Rings 221
14.2 Polynomial Congruences 226
14.3 Polynomial Congruence Rings 230
14.4 Equations and Congruences with Polynomial Unknowns 233
14.5 Polynomial Congruence Fields 236
Part III All Together Now
15 Euclidean Rings 241
15.1 Factoring Elements in Rings 241
15.2 Euclidean Rings 245
15.3 Unique Factorization 249
16
The Ring of Gaussian Integers 255
16.1 The Irreducible Gaussian Integers 255
16.2 Gaussian Congruence Rings 259
16.3 Fermat's Theorem 262
17 Finite Fields 267
17.1 Primitive Roots 267
17.2 Quadratic Reciprocity 271
17.3 Classification 277