Cover

Title page

Contents

Preface

Chapter 1. Preliminaries

1.1. Basic Properties of Finite Fields

1.2. Partially Ordered Sets and the Möbius Function

Exercises

Chapter 2. Polynomials over Finite Fields

2.1. Number of Irreducible Polynomials

2.2. Berlekamp’s Factorization Algorithm

2.3. Functions from 𝔽_{𝕢}ⁿ to 𝔽_{𝕢}

2.4. Permutation Polynomials

2.5. Linearized Polynomials

2.6. Payne’s Theorem

Exercises

Chapter 3. Gauss Sums

3.1. Characters of Finite Abelian Groups

3.2. Gauss Sums

3.3. The Davenport-Hasse Theorem

3.4. The Gauss Quadratic Sum

Exercises

Chapter 4. Algebraic Number Theory

4.1. Number Fields

4.2. Ramification and Degree

4.3. Extensions of Number Fields

4.4. Factorization of Primes

4.5. Cyclotomic Fields

4.6. Stickelberger’s Congruence

Exercises

Chapter 5. Zeros of Polynomials over Finite Fields

5.1. Ax’s Theorem

5.2. Katz’s Theorem

5.3. Bounds on the Number of Zeros of Polynomials

5.4. Bounds Derived from Function Fields

Exercises

Chapter 6. Classical Groups

6.1. The General Linear Group and Its Related Groups

6.2. Simplicity of 𝑃𝑆𝐿(𝑛,𝐹)

6.3. Conjugacy Classes of 𝐺𝐿(𝑛,𝔽_{𝕢})

6.4. Conjugacy Classes of 𝐴𝐺𝐿(𝑛,𝔽_{𝕢})

6.5. Bilinear Forms, Hermitian Forms, and Quadratic Forms

6.6. Groups of Spaces Equipped with Forms

Exercises

Bibliography

List of Notation

Index

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