Excursions in Number Theory, Algebra, and Analysis

This textbook originates from a course taught by the late Ken Ireland in 1972. Designed to explore the theoretical underpinnings of undergraduate mathematics, the course focused on interrelationships and hands-on experience. Readers of this textbook will be taken on a modern rendering of Ireland’s path of discovery, consisting of excursions into number theory, algebra, and analysis. Replete with surprising connections, deep insights, and brilliantly curated invitations to try problems at just the right moment, this journey weaves a rich body of knowledge that is ideal for those going on to study or teach mathematics.

A pool of 200 ‘Dialing In’ problems opens the book, providing fuel for active enquiry throughout a course. The following chapters develop theory to illuminate the observations and roadblocks encountered in the problems, situating them in the broader mathematical landscape. Topics cover polygons and modular arithmetic; the fundamental theorems of arithmetic and algebra; irrational, algebraic and transcendental numbers; and Fourier series and Gauss sums. A lively accompaniment of examples, exercises, historical anecdotes, and asides adds motivation and context to the theory. Return trips to the Dialing In problems are encouraged, offering opportunities to put theory into practice and make lasting connections along the way.

Excursions in Number Theory, Algebra, and Analysis invites readers on a journey as important as the destination. Suitable for a senior capstone, professional development for practicing teachers, or independent reading, this textbook offers insights and skills valuable to math majors and high school teachers alike. A background in real analysis and abstract algebra is assumed, though the most important prerequisite is a willingness to put pen to paper and do some mathematics.

Kenneth Ireland [1937–1991] was Professor of Mathematics at the University of New Brunswick, Canada. Before UNB, he held positions at Johns Hopkins, Brandeis, Brown, Bowdoin, and a visiting membership in the Institute for Advanced Study. Known for his broad interests and active involvement in all areas of departmental life, he had a lasting impact on generations of students as a teacher and mentor. His widely acclaimed book with Michael Rosen, A Classical Introduction to Modern Number Theory, appears in the series Graduate Texts in Mathematics.

Al Cuoco is a Distinguished Scholar at Education Development Center in Massachusetts. Before EDC, he taught high school mathematics for 24 years to a wide range of students in the Woburn, Massachusetts public schools. At EDC, he works in curriculum development, professional development, and education policy, and for over three decades, he has worked with Glenn Stevens in the PROMYS for Teachers program atBoston University. His mathematical interests and research publications lie in algebraic number theory.