ISBN13: 978-0387955872

ISBN10: 0387955879

Cover type:

Edition/Copyright: 02

Publisher: Springer-Verlag New York

Published: 2002

International: No

ISBN10: 0387955879

Cover type:

Edition/Copyright: 02

Publisher: Springer-Verlag New York

Published: 2002

International: No

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This book is a concise introduction to number theory and some related algebra, with an emphasis on solving equations in integers. Finding integer solutions led to two fundamental ideas of number theory in ancient times - the Euclidean algorithm and unique prime factorization - and in modern times to two fundamental ideas of algebra - rings and ideals. The development of these ideas, and the transition from ancient to modern, is the main theme of the book. The historical development has been followed where it helps to motivate the introduction of new concepts, but modern proofs have been used where they are simpler, more natural, or more interesting. These include some that have not yet appeared in textbooks, such as a treatment of the Pell equation using Conway's theory of quadratic forms. Also, this is the only elementary number theory book that includes significant applications of ideal theory. It is clearly written, well illustrated, and supplied with carefully designed exercises, making it a pleasure to use as an undergraduate textbook or for independent study.

Natural numbers and integers

The Euclidean algorithm

Congruence arithmetic

The RSA cryptosystem

The Pell equation

The Gaussian Integers

Quadratic integers

The four square theorem

Quadratic reciprocity

Rings

Ideals

Prime ideals

Bibliography

Index

ISBN10: 0387955879

Cover type:

Edition/Copyright: 02

Publisher: Springer-Verlag New York

Published: 2002

International: No

This book is a concise introduction to number theory and some related algebra, with an emphasis on solving equations in integers. Finding integer solutions led to two fundamental ideas of number theory in ancient times - the Euclidean algorithm and unique prime factorization - and in modern times to two fundamental ideas of algebra - rings and ideals. The development of these ideas, and the transition from ancient to modern, is the main theme of the book. The historical development has been followed where it helps to motivate the introduction of new concepts, but modern proofs have been used where they are simpler, more natural, or more interesting. These include some that have not yet appeared in textbooks, such as a treatment of the Pell equation using Conway's theory of quadratic forms. Also, this is the only elementary number theory book that includes significant applications of ideal theory. It is clearly written, well illustrated, and supplied with carefully designed exercises, making it a pleasure to use as an undergraduate textbook or for independent study.

Table of Contents

The Euclidean algorithm

Congruence arithmetic

The RSA cryptosystem

The Pell equation

The Gaussian Integers

Quadratic integers

The four square theorem

Quadratic reciprocity

Rings

Ideals

Prime ideals

Bibliography

Index

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