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Excursions in Number Theory, Algebra, and Analysis

Kenneth Ireland and Al Cuoco
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
John D. Cook
, on
Excursions in Number Theory, Algebra, and Analysis by Kenneth Ireland and Al Cuoco differs from most mathematics text books in two ways: its structure and its breadth.
The book is in the series Undergraduate Texts in Mathematics. It contains mostly undergraduate-level material, but the breadth and style would be appropriate for a graduate-level course or an undergraduate capstone course, a course in which the boundaries between subject classifications are broken down.
Excursions is very much about doing mathematics, not just conveying well-known mathematical theories. It is written closer to the way math is done in practice than the way math is typically presented: Math is discovered inductively but usually taught more deductively, a legacy of Euclid I suppose.
Excursions starts with a list of problems it calls “Dialing In problems.” The text states that these problems are “designed to help you ‘dial in’ to several mathematical structures and theories” and exhorts the reader to take these problems seriously.  These problems make up the heart of what you’ll learn from this book. The chapters exist to support your work on the Dialing In problems.
True to its title, Excursions delves into number theory, algebra, and analysis. It could be used as an algebra textbook, easing into standard algebra topics starting from motivating problems in number theory, following the authors’ philosophy of ”experience before formality.” Following the experience gained from following the book, an instructor could then supplement with more of the usual formality.
Excursions also delves into analysis, but would not make a suitable analysis textbook per se. The analysis portions of the book are more along the lines of analytic number theory than more general analysis. The main benefit of the book from an analyst’s viewpoint is seeing analysis arise naturally in what for students may be an unexpected context.
A word about the authorship of the book may be in order. The late Kenneth Ireland (1937–1991) is listed as the first author, though one may say this is a book by Al Cuoco, based (perhaps heavily) on notes by his former teacher Kenneth Ireland. This distinction makes it easier to understand some of the first person pronouns in the book.
Rather than being a book that one reads from cover to cover, Excursions is a curated collection problems followed by expository material aimed at providing background material useful for solving these problems. I imagine it would be a great experience to have a course taught out of this book. The second author clearly enjoyed the experience of studying this material under the guidance of the first author and wanted to make that experience available to others.
John D. Cook is a consultant working in applied mathematics.