**Best Reference Books on Finite Fields**, which are used by students of top universities, and colleges. This will help you choose the right book depending on if you are a beginner or an expert. Here is the complete list of

**Finite Fields Books**with their authors, publishers, and an unbiased review of them as well as links to the Amazon website to directly purchase them. If permissible, you can also download the free PDF books on Finite Fields below.

## 1. Computations on Finite Fields

1."Finite Fields: Theory and Computation: The Meeting Point of Number Theory, Computer Science, Coding Theory and Cryptography (Mathematics and Its Applications)" by Igor Shparlinski
“Finite Fields: Theory and Computation: The Meeting Point of Number Theory, Computer Science, Coding Theory and Cryptography (Mathematics and Its Applications)” Book Review: This book provides a comprehensive analysis of various computational and algorithmic problems in finite fields with applications in computer science, cryptography, numerical methods, coding theory, and more. The book delves into topics such as polynomial factorization, identifying irreducible and primitive polynomials, distribution of primitive points on elliptic curves, and constructing various types of bases, all with detailed explanations. It also contains important results from international conferences and journals, as well as theoretical problems and numerical examples. Additionally, the book covers recent advances in the field of congruences, including optimal coefficients, congruential pseudo-random number generators, and modular arithmetic.
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2."Primality Testing and Abelian Varieties Over Finite Fields (Lecture Notes in Mathematics)" by Leonard M Adleman and Ming-Deh A Huang
“Primality Testing and Abelian Varieties Over Finite Fields (Lecture Notes in Mathematics)” Book Review: The target audience for this book is researchers and graduate students studying number theory and computer science. It discusses topics such as the random polynomial time algorithm and the theory of two-dimensional abelian developments. In addition, the book provides various approaches to solving numerical problems related to arithmetic algebraic geometry, algebraic number theory, and analytic number theory.
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3."Finite Fields and Applications" by Gary L Mullen and Alain Poli
“Finite Fields and Applications” Book Review: This book is a comprehensive and well-written book that provides a thorough introduction to the theory of finite fields and their applications.The book is divided into several sections that cover topics such as the basic properties of finite fields, polynomial factorization algorithms, cryptography, coding theory, and more. Each chapter includes numerous examples, exercises, and references to other texts, making it easy for readers to follow along and apply the concepts they learn. It contains the insights of the 7th international conference of finite fields and applications.
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4."Quantum Information Processing with Finite Resources: Mathematical Foundations (SpringerBriefs in Mathematical Physics)" by Marco Tomamichel
“Quantum Information Processing with Finite Resources: Mathematical Foundations (SpringerBriefs in Mathematical Physics)” Book Review: This book is intended for graduate students in physics and IT seeking a comprehensive understanding of the applications of statistical theory and cryptography. In addition to covering topics such as the formalism of quantum mechanics, metrics for quantum states, and norms, it also explains the quantum generalizations of Rényi divergence and conditional entropy.
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## 2. Finite Fields and their Applications

1."Finite Fields" by Rudolf Lidl and Harald Niederreiter
Book Review: The book covers a wide range of topics such as combinatorics, coding theory, cryptology, and switching circuits. It offers an introduction to the theoretical foundations of these areas, emphasizing their applications. In addition, the book includes chapters on finite field applications, information theory, algebraic coding theory, cryptology, finite geometries, combinatorics, and pseudo-random sequences. With worked examples and exercises, this book serves as an excellent graduate-level textbook.
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2."Introduction to Finite Fields and their Applications" by R Lidl and H Niederreiter
Book Review: The book offers a comprehensive coverage of diverse subjects such as combinatorics, coding theory, cryptology, switching circuits, and more. It provides an overview of the theoretical aspects of these subjects while emphasizing their relevance to real-world applications. Moreover, the book includes chapters on finite field applications, information theory, algebraic coding theory, cryptology, finite geometries, combinatorics, and pseudo-random sequences. With its detailed worked examples and exercises, this book is a valuable resource for graduate students.
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3."Finite Fields and Their Applications: Character Sums and Polynomials" by Pascale Charpin and Alexander Pott
“Finite Fields and their Applications: Character Sums and Polynomials” Book Review: This book discuss the recent research on sequences, polynomials, and character sums. It begins with an overview of polynomials over finite fields and the construction and analysis of sequences using characters and character sum bounds. In Chapter 8, the book explores the iterations of rational functions, which can be used to generate pseudorandom numbers for Monte Carlo methods. Additionally, the book examines the applications of quasirandom points in numerical integration using quasi-Monte Carlo methods and simulation. Furthermore, it discusses problems related to coding theory, which are studied through the lenses of finite geometry and additive combinatorics.
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4."Finite Fields and Applications: 7th International Conference, Fq7 Toulouse, France, May 5-9, 2003 Revised Papers" by Mullen Poli
“Finite Fields and Applications: 7th International Conference, Fq7, Toulouse, France, May 5-9, 2003, Revised Paper” Book Review: The selected papers in this book are from the 7th International Conference on Finite Fields and Applications, Fq7, held in Toulouse, France in May 2003. It covers topics like Weierstrass semigroups, Galois rings, hyperelliptic curves, polynomial irreducibility, pseudorandom number sequences, permutation polynomials, and random polynomials in the initial chapters. The following chapters discuss matrices, function fields, ramified towers, BCH codes, cyclic codes, primitive polynomials, covering sequences, and cyclic decompositions.
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5."Finite Fields with Applications to Coding Theory, Cryptography and Related Areas" by Gary L Mullen and Henning Stichtenoth
“Finite Fields with Applications to Coding Theory, Cryptography and Related Areas” Book Review: The proceedings of the Sixth International Conference on Finite Fields and Applications, Fq6, held in Oaxaca, Mexico, from 22-26 May 2001 are included in this book. It provides a comprehensive discussion of the basic principles of finite fields and their practical applications. The book thoroughly explores the significance of finite fields in diverse areas such as discrete mathematics, cryptography, coding theory, design theory, finite & algebraic geometry, scientific computation, and more.
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6."Finite Fields and Applications" by H Niederreiter and Dieter Jungnickel
“Finite Fields and Applications” Book Review: This book presents the proceedings of the Fifth International Conference on Finite Fields and Applications Fq5, which took place at the University of Augsburg, Germany, from August 2-6, 1999. It features a collection of up-to-date research articles that explore both the theoretical and practical aspects of finite fields. The book covers a diverse range of applications, spanning from combinatorial design theory, finite geometries, and algebraic geometry to coding theory, cryptology, and scientific computing, all of which are analyzed in great detail.
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7."Rational Points on Curves Over Finite Fields: Theory And Applications" by J W S Cassels and Harald Niederreiter
“Rational Points on Curves Over Finite Fields: Theory And Applications” Book Review: This book provides an in-depth examination of rational points on algebraic curves over finite fields. It provides a comprehensive overview of the theoretical work on algebraic curves with many rational points. In addition, it explores the practical applications of this theory, including algebraic coding theory in information theory and the construction of low-discrepancy sequences in computational mathematics. The authors present the first organized treatment of this material in the book, making it valuable for both algebraic geometers and coding theorists.
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8."Handbook of Finite Fields" by Mullen Panario
“Handbook of Finite Fields” Book Review: This book comprehensively covers the foundational concepts and recent research conducted in finite fields, in a self-contained manner. The historical background of finite fields is presented initially, followed by detailed coverage of their theoretical properties including polynomials, algorithms, curves, special functions, sequences, and computational aspects. The book also includes a comprehensive index and 3,000 references for thorough research. Various mathematical and practical applications of finite fields in quantum information theory, algebraic coding theory, cryptographic systems, combinatorics, engineering, and biology are thoroughly described and analyzed.
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