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John Stillwell
Springer

Elements of Number Theory

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AuthorJohn Stillwell
PublisherSpringer
PublishedMay 30 2022
Book FormatHardcover
ISBN-139780387955872
Dimensions9.3 X 6.4 X 0.9 inches
Weight 1.2 pounds
BISAC
CategoriesMathematics > Number Theory
LanguageEnglish

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Book Description

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.

Author Bio

John Stillwell was born in Melbourne, Australia, and taught at Monash University from 1970 until 2001, before moving to USF in 2002.

He was an invited speaker at the International Congress of Mathematicians in 1994, and his mathematical writing has been honored with the Chauvenet Prize of the Mathematical Association of America in 2005 and the book award of the Association of Jesuit Colleges and Universities in 2009.

Among his best-known books are Mathematics and Its History (3rd edition, 2010) and Yearning for the Impossible (winner of the AJCU book award in 2009).

His interests are history of mathematics in the 19th and 20th centuries, number theory, geometry, algebra, topology, foundations of mathematics.

In Australia during spring and summer.

Source: University of San Francisco

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