Finite fields, with applications to combinatorics
著者
書誌事項
Finite fields, with applications to combinatorics
（Student mathematical library, v. 99）
American Mathematical Society, c2022
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注記
Includes bibliographical references (p. 165-167) and index
内容説明・目次
内容説明
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.
目次
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
Bibliography
Index
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