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Cover: 9781441930668 | Elements of Number Theory | John Stillwell | Taschenbuch | Paperback
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Elements of Number Theory
Taschenbuch von John Stillwell
Sprache: Englisch

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Beschreibung
This book is intended to complement my Elements oi Algebra, and it is similarly motivated by the problem of solving polynomial equations. However, it is independent of the algebra book, and probably easier. In Elements oi Algebra we sought solution by radicals, and this led to the concepts of fields and groups and their fusion in the celebrated theory of Galois. In the present book we seek integer solutions, and this leads to the concepts of rings and ideals which merge in the equally celebrated theory of ideals due to Kummer and Dedekind. Solving equations in integers is the central problem of number theory, so this book is truly a number theory book, with most of the results found in standard number theory courses. However, numbers are best understood through their algebraic structure, and the necessary algebraic concepts­ rings and ideals-have no better motivation than number theory. The first nontrivial examples of rings appear in the number theory of Euler and Gauss. The concept of ideal-today as routine in ring the­ ory as the concept of normal subgroup is in group theory-also emerged from number theory, and in quite heroic fashion. Faced with failure of unique prime factorization in the arithmetic of certain generalized "inte­ gers" , Kummer created in the 1840s a new kind of number to overcome the difficulty. He called them "ideal numbers" because he did not know exactly what they were, though he knew how they behaved.
This book is intended to complement my Elements oi Algebra, and it is similarly motivated by the problem of solving polynomial equations. However, it is independent of the algebra book, and probably easier. In Elements oi Algebra we sought solution by radicals, and this led to the concepts of fields and groups and their fusion in the celebrated theory of Galois. In the present book we seek integer solutions, and this leads to the concepts of rings and ideals which merge in the equally celebrated theory of ideals due to Kummer and Dedekind. Solving equations in integers is the central problem of number theory, so this book is truly a number theory book, with most of the results found in standard number theory courses. However, numbers are best understood through their algebraic structure, and the necessary algebraic concepts­ rings and ideals-have no better motivation than number theory. The first nontrivial examples of rings appear in the number theory of Euler and Gauss. The concept of ideal-today as routine in ring the­ ory as the concept of normal subgroup is in group theory-also emerged from number theory, and in quite heroic fashion. Faced with failure of unique prime factorization in the arithmetic of certain generalized "inte­ gers" , Kummer created in the 1840s a new kind of number to overcome the difficulty. He called them "ideal numbers" because he did not know exactly what they were, though he knew how they behaved.
Zusammenfassung
This book is a concise introduction to number theory and some related
algebra, with an emphasis on solving equations in integers. 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.
Inhaltsverzeichnis
* Preface * 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
Details
Erscheinungsjahr: 2010
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 268
Reihe: Undergraduate Texts in Mathematics
Inhalt: xii
256 S.
2 s/w Illustr.
ISBN-13: 9781441930668
ISBN-10: 1441930663
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Stillwell, John
Auflage: Softcover reprint of hardcover 1st ed. 2003
Hersteller: Springer US
Springer New York
Undergraduate Texts in Mathematics
Maße: 235 x 155 x 15 mm
Von/Mit: John Stillwell
Erscheinungsdatum: 03.12.2010
Gewicht: 0,411 kg
preigu-id: 107252898
Zusammenfassung
This book is a concise introduction to number theory and some related
algebra, with an emphasis on solving equations in integers. 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.
Inhaltsverzeichnis
* Preface * 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
Details
Erscheinungsjahr: 2010
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 268
Reihe: Undergraduate Texts in Mathematics
Inhalt: xii
256 S.
2 s/w Illustr.
ISBN-13: 9781441930668
ISBN-10: 1441930663
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Stillwell, John
Auflage: Softcover reprint of hardcover 1st ed. 2003
Hersteller: Springer US
Springer New York
Undergraduate Texts in Mathematics
Maße: 235 x 155 x 15 mm
Von/Mit: John Stillwell
Erscheinungsdatum: 03.12.2010
Gewicht: 0,411 kg
preigu-id: 107252898
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