Elements of Number Theory

John Stillwell (Autor)

Buch | Softcover

256 Seiten

2010 | Softcover reprint of hardcover 1st ed. 2003
Springer-Verlag New York Inc.
978-1-4419-3066-8 (ISBN)

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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.

1 Natural numbers and integers.- 2 The Euclidean algorithm.- 3 Congruence arithmetic.- 4 The RSA cryptosystem.- 5 The Pell equation.- 6 The Gaussian integers.- 7 Quadratic integers.- 8 The four square theorem.- 9 Quadratic reciprocity.- 10 Rings.- 11 Ideals.- 12 Prime ideals.

Erscheint lt. Verlag	3.12.2010
Reihe/Serie	Undergraduate Texts in Mathematics
Zusatzinfo	XII, 256 p.
Verlagsort	New York, NY
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
ISBN-10	1-4419-3066-3 / 1441930663
ISBN-13	978-1-4419-3066-8 / 9781441930668
Zustand	Neuware