Lectures on Finite Fields

Cover......Page 1
Title page......Page 2
Contents......Page 6
Preface......Page 8
1.1. Basic Properties of Finite Fields......Page 12
1.2. Partially Ordered Sets and the Möbius Function......Page 23
Exercises......Page 28
2.1. Number of Irreducible Polynomials......Page 34
2.2. Berlekamp’s Factorization Algorithm......Page 37
2.3. Functions from ��_{��}ⁿ to ��_{��}......Page 43
2.4. Permutation Polynomials......Page 51
2.5. Linearized Polynomials......Page 57
2.6. Payne’s Theorem......Page 61
Exercises......Page 65
3.1. Characters of Finite Abelian Groups......Page 68
3.2. Gauss Sums......Page 75
3.3. The Davenport-Hasse Theorem......Page 78
3.4. The Gauss Quadratic Sum......Page 81
Exercises......Page 84
4.1. Number Fields......Page 88
4.2. Ramification and Degree......Page 98
4.3. Extensions of Number Fields......Page 100
4.4. Factorization of Primes......Page 106
4.5. Cyclotomic Fields......Page 107
4.6. Stickelberger’s Congruence......Page 113
Exercises......Page 116
5.1. Ax’s Theorem......Page 122
5.2. Katz’s Theorem......Page 127
5.3. Bounds on the Number of Zeros of Polynomials......Page 130
5.4. Bounds Derived from Function Fields......Page 138
Exercises......Page 150
Chapter 6. Classical Groups......Page 154
6.1. The General Linear Group and Its Related Groups......Page 155
6.2. Simplicity of ������(��,��)......Page 157
6.3. Conjugacy Classes of ����(��,��_{��})......Page 164
6.4. Conjugacy Classes of ������(��,��_{��})......Page 171
6.5. Bilinear Forms, Hermitian Forms, and Quadratic Forms......Page 183
6.6. Groups of Spaces Equipped with Forms......Page 203
Exercises......Page 226
Bibliography......Page 232
List of Notation......Page 234
Index......Page 238
Back Cover......Page 242