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Inhaltsangabe
Solutions of equations in integers is the central problem of number theory and is the focus of this book. The amount of material is suitable for a one-semester course. The author has tried to avoid the ad hoc proofs in favor of unifying ideas that work in many situations. There are exercises at the end of almost every section, so that each new idea or proof receives immediate reinforcement.
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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.
John Stillwell is Professor of Mathematics at the University of San Francisco. He is the author of several highly regarded books published by Springer-Verlag, including Mathematics and Its History (Second Edition 2001), Numbers and Geometry (1997) and Elements of Algebra (1994).
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- VerlagSpringer
- Erscheinungsdatum2002
- ISBN 10 0387955879
- ISBN 13 9780387955872
- EinbandTapa dura
- SpracheEnglisch
- Anzahl der Seiten272
- Kontakt zum HerstellerNicht verfügbar
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Elements of Number Theory (Undergraduate Texts in Mathematics)
Verlag:
Springer (edition 2003), 2002
ISBN 10: 0387955879
ISBN 13: 9780387955872
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Éléments of number theory - John Stillwell
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Elements of Number Theory
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Buch. Zustand: Neu. This item is printed on demand - it takes 3-4 days longer - Neuware -Solutions of equations in integers is the central problem of number theory and is the focus of this book. The amount of material is suitable for a one-semester course. The author has tried to avoid the ad hoc proofs in favor of unifying ideas that work in many situations. There are exercises at the end of almost every section, so that each new idea or proof receives immediate reinforcement. 272 pp. Englisch. Bestandsnummer des Verkäufers 9780387955872
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Elements of Number Theory
Verlag:
Springer New York, Springer New York, 2002
ISBN 10: 0387955879
ISBN 13: 9780387955872
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Buch. Zustand: Neu. Druck auf Anfrage Neuware - Printed after ordering - 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. Bestandsnummer des Verkäufers 9780387955872
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