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Finite Fields, with Applications to Combinatorics
 
Kannan Soundararajan Stanford University, Stanford, CA
Softcover ISBN:  978-1-4704-6930-6
Product Code:  STML/99
List Price: $59.00
Individual Price: $47.20
Sale Price: $38.35
eBook ISBN:  978-1-4704-7242-9
Product Code:  STML/99.E
List Price: $59.00
Individual Price: $47.20
Sale Price: $38.35
Softcover ISBN:  978-1-4704-6930-6
eBook: ISBN:  978-1-4704-7242-9
Product Code:  STML/99.B
List Price: $118.00 $88.50
Sale Price: $76.70 $57.53
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Finite Fields, with Applications to Combinatorics
Kannan Soundararajan Stanford University, Stanford, CA
Softcover ISBN:  978-1-4704-6930-6
Product Code:  STML/99
List Price: $59.00
Individual Price: $47.20
Sale Price: $38.35
eBook ISBN:  978-1-4704-7242-9
Product Code:  STML/99.E
List Price: $59.00
Individual Price: $47.20
Sale Price: $38.35
Softcover ISBN:  978-1-4704-6930-6
eBook ISBN:  978-1-4704-7242-9
Product Code:  STML/99.B
List Price: $118.00 $88.50
Sale Price: $76.70 $57.53
  • Book Details
     
     
    Student Mathematical Library
    Volume: 992022; 170 pp
    MSC: 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 pre-requisites 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.

    Readership

    Undergraduate 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
  • Reviews
     
     
    • Throughout this book, many interesting examples and exercises are included in each chapter, which in my opinion, makes of this book a very good one.

      Moisés R. Delgado (San Juan), ZbMATHOpen
    • The writing is very clear, and there are abundant cross-references 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Accessibility – to request an alternate format of an AMS title
Volume: 992022; 170 pp
MSC: 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 pre-requisites 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.

Readership

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
  • Throughout this book, many interesting examples and exercises are included in each chapter, which in my opinion, makes of this book a very good one.

    Moisés R. Delgado (San Juan), ZbMATHOpen
  • The writing is very clear, and there are abundant cross-references 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
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.