Softcover ISBN:  9781470443993 
Product Code:  STML/88 
List Price:  $55.00 
Individual Price:  $44.00 
MAA Member Price:  $44.00 
Electronic ISBN:  9781470452612 
Product Code:  STML/88.E 
List Price:  $55.00 
Individual Price:  $44.00 
MAA Member Price:  $44.00 

Book DetailsStudent Mathematical LibraryVolume: 88; 2019; 239 ppMSC: Primary 11; 12;
Hilbert's tenth problem is one of 23 problems proposed by David Hilbert in 1900 at the International Congress of Mathematicians in Paris. These problems gave focus for the exponential development of mathematical thought over the following century. The tenth problem asked for a general algorithm to determine if a given Diophantine equation has a solution in integers. It was finally resolved in a series of papers written by Julia Robinson, Martin Davis, Hilary Putnam, and finally Yuri Matiyasevich in 1970. They showed that no such algorithm exists.
This book is an exposition of this remarkable achievement. Often, the solution to a famous problem involves formidable background. Surprisingly, the solution of Hilbert's tenth problem does not. What is needed is only some elementary number theory and rudimentary logic. In this book, the authors present the complete proof along with the romantic history that goes with it. Along the way, the reader is introduced to Cantor's transfinite numbers, axiomatic set theory, Turing machines, and Gödel's incompleteness theorems.
Copious exercises are included at the end of each chapter to guide the student gently on this ascent. For the advanced student, the final chapter highlights recent developments and suggests future directions. The book is suitable for undergraduates and graduate students. It is essentially selfcontained.
Cover image by Jesse Jacobs.ReadershipUndergraduate and graduate students and researchers interested in number theory and logic.

Table of Contents

Chapters

Introduction

Cantor and infinity

Axiomatic set theory

Elementary number theory

Computability and provability

Hilbert’s tenth problem

Applications of Hilbert’s tenth problem

Hilbert’s tenth problem over number fields

Background material


Additional Material

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Hilbert's tenth problem is one of 23 problems proposed by David Hilbert in 1900 at the International Congress of Mathematicians in Paris. These problems gave focus for the exponential development of mathematical thought over the following century. The tenth problem asked for a general algorithm to determine if a given Diophantine equation has a solution in integers. It was finally resolved in a series of papers written by Julia Robinson, Martin Davis, Hilary Putnam, and finally Yuri Matiyasevich in 1970. They showed that no such algorithm exists.
This book is an exposition of this remarkable achievement. Often, the solution to a famous problem involves formidable background. Surprisingly, the solution of Hilbert's tenth problem does not. What is needed is only some elementary number theory and rudimentary logic. In this book, the authors present the complete proof along with the romantic history that goes with it. Along the way, the reader is introduced to Cantor's transfinite numbers, axiomatic set theory, Turing machines, and Gödel's incompleteness theorems.
Copious exercises are included at the end of each chapter to guide the student gently on this ascent. For the advanced student, the final chapter highlights recent developments and suggests future directions. The book is suitable for undergraduates and graduate students. It is essentially selfcontained.
Cover image by Jesse Jacobs.
Undergraduate and graduate students and researchers interested in number theory and logic.

Chapters

Introduction

Cantor and infinity

Axiomatic set theory

Elementary number theory

Computability and provability

Hilbert’s tenth problem

Applications of Hilbert’s tenth problem

Hilbert’s tenth problem over number fields

Background material