Softcover ISBN:  9781470469306 
Product Code:  STML/99 
List Price:  $59.00 
Individual Price:  $47.20 
Electronic ISBN:  9781470472429 
Product Code:  STML/99.E 
List Price:  $59.00 
Individual Price:  $47.20 

Book DetailsStudent Mathematical LibraryVolume: 99; 2022; 170 ppMSC: Primary 11; 05; 12;
This book uses finite field theory as a hook to introduce the reader to a range of ideas from algebra and number theory. It constructs all finite fields from scratch and shows that they are unique up to isomorphism. As a payoff, several combinatorial applications of finite fields are given: Sidon sets and perfect difference sets, de Bruijn sequences and a magic trick of Persi Diaconis, and the polynomial time algorithm for primality testing due to Agrawal, Kayal and Saxena.
The book forms the basis for a one term intensive course with students meeting weekly for multiple lectures and a discussion session. Readers can expect to develop familiarity with ideas in algebra (groups, rings and fields), and elementary number theory, which would help with later classes where these are developed in greater detail. And they will enjoy seeing the AKS primality test application tying together the many disparate topics from the book. The prerequisites for reading this book are minimal: familiarity with proof writing, some linear algebra, and one variable calculus is assumed. This book is aimed at incoming undergraduate students with a strong interest in mathematics or computer science.ReadershipUndergraduate students interested in finite fields and combinatorics.

Table of Contents

Chapters

Primes and factorization

Primes in the integers

Congruences in rings

Primes in polynomial rings: Constructing finite fields

The additive and multiplicative structures of finite fields

Understanding the structures of $\mathbb {Z}/n\mathbb {Z}$

Combinatorial applications of finite fields

The AKS primality test

Synopsis of finite fields


Additional Material

Reviews

The writing is very clear, and there are abundant crossreferences and a good index in case you want to start in the middle of things rather than reading straight through. In particular the book is valuable if you already know about finite fields but would like to see some interesting applications. As abstract algebra texts go, this treatment is very concrete with lots of specific examples. The book has a strong number theory flavor and brings out how these abstract structures generalize the integers.
Allen Stenger, MAA Reviews


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This book uses finite field theory as a hook to introduce the reader to a range of ideas from algebra and number theory. It constructs all finite fields from scratch and shows that they are unique up to isomorphism. As a payoff, several combinatorial applications of finite fields are given: Sidon sets and perfect difference sets, de Bruijn sequences and a magic trick of Persi Diaconis, and the polynomial time algorithm for primality testing due to Agrawal, Kayal and Saxena.
The book forms the basis for a one term intensive course with students meeting weekly for multiple lectures and a discussion session. Readers can expect to develop familiarity with ideas in algebra (groups, rings and fields), and elementary number theory, which would help with later classes where these are developed in greater detail. And they will enjoy seeing the AKS primality test application tying together the many disparate topics from the book. The prerequisites for reading this book are minimal: familiarity with proof writing, some linear algebra, and one variable calculus is assumed. This book is aimed at incoming undergraduate students with a strong interest in mathematics or computer science.
Undergraduate students interested in finite fields and combinatorics.

Chapters

Primes and factorization

Primes in the integers

Congruences in rings

Primes in polynomial rings: Constructing finite fields

The additive and multiplicative structures of finite fields

Understanding the structures of $\mathbb {Z}/n\mathbb {Z}$

Combinatorial applications of finite fields

The AKS primality test

Synopsis of finite fields

The writing is very clear, and there are abundant crossreferences and a good index in case you want to start in the middle of things rather than reading straight through. In particular the book is valuable if you already know about finite fields but would like to see some interesting applications. As abstract algebra texts go, this treatment is very concrete with lots of specific examples. The book has a strong number theory flavor and brings out how these abstract structures generalize the integers.
Allen Stenger, MAA Reviews