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This book introduces algebraic number theory through the problem of generalizing 'unique prime factorization' from ordinary integers to more general domains. Solving polynomial equations in integers leads naturally to these domains, but unique prime factorization may be lost in the process. To restore it, we need Dedekind's concept of ideals. However, one still needs the supporting concepts of algebraic number field and algebraic integer, and the supporting theory of rings, vector spaces, and modules. It was left to Emmy Noether to encapsulate the properties of rings that make unique prime factorization possible, in what we now call Dedekind rings. The book develops the theory of these concepts, following their history, motivating each conceptual step by pointing to its origins, and focusing on the goal of unique prime factorization with a minimum of distraction or prerequisites. This makes a self-contained easy-to-read book, short enough for a one-semester course.

This Element aims to present an outline of mathematics and its history, with particular emphasis on events that shook up its philosophy. It ranges from the discovery of irrational numbers in ancient Greece to the nineteenth- and twentieth-century discoveries on the nature of infinity and proof. Recurring themes are intuition and logic, meaning and existence, and the discrete and the continuous. These themes have evolved under the influence of new mathematical discoveries and the story of their evolution is, to a large extent, the story of philosophy of mathematics.

This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathema

This text plugs a gap in the standard curriculum by linking set theory with analysis. It features a distinctive, detailed treatment of the real numbers system, and combines an introduction to set theo

From a review of the second edition:"This book covers many interesting topics not usually covered in a present day undergraduate course, as well as certain basic topics such as the development of the

From a review of the second edition:"This book covers many interesting topics not usually covered in a present day undergraduate course, as well as certain basic topics such as the development of the

A beautiful and relatively elementary account of a part of mathematics where three main fields - algebra, analysis and geometry - meet. The book provides a broad view of these subjects at the level of

The first book surveying the history and ideas behind reverse mathematicsReverse mathematics is a new field that seeks to find the axioms needed to prove given theorems. In Reverse Mathematics, John S

Elements of Mathematics takes readers on a fascinating tour that begins in elementary mathematics--but, as John Stillwell shows, this subject is not as elementary or straightforward as one might think

Elements of Mathematics takes readers on a fascinating tour that begins in elementary mathematics--but, as John Stillwell shows, this subject is not as elementary or straightforward as one might think

In this new textbook, acclaimed author John Stillwell presents a lucid introduction to Lie theory suitable for junior and senior level undergraduates. In order to achieve this, he focuses on the so-ca

In this new textbook, acclaimed author John Stillwell presents a lucid introduction to Lie theory suitable for junior and senior level undergraduates. In order to achieve this, he focuses on the so-ca

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 th

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 th

This book is a concise, self-contained introduction to abstract algebra that stresses its unifying role in geometry and number theory. Classical results in these fields, such as the straightedge-and-c

This is a well-balanced introduction to topology that stresses geometric aspects. Focusing on historical background and visual interpretation of results, it emphasizes spaces with few dimensions, wher

While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. An