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An open door to number theory
Campbell D., AMERICAN MATHEMATICAL SOCIETY, Washington, DC, 2018. 283 pp. Type: Book (978-1-470443-48-1)
Date Reviewed: Feb 19 2019

In 1993, my daughter showed me the New York Times article that announced Andrew Wiles’s 100-plus-page proof (to be successfully corrected two years later) of Fermat’s last theorem (FLT) [1]. Wiles’s “20th-century” proof comprised excursions into algebraic geometry and other territory that did not at all resemble the seemingly clean and pure realm of number theory “proper.” The book captures the best spirit of contemporary mathematics at large, and achieves cohesion among the natural numbers ℕ = {1, 2, 3, ...}; the whole numbers 𝕎 = {0, 1, 2, ...}; the integers ℤ; the rational numbers ℚ; the algebraic numbers 𝔸; the real numbers ℝ; and the complex numbers ℂ.

Chapter 1, “Integers”: this 70-page chapter, the first of six, already gives strong confirmation of the author’s self-description of “mathematical generalist,” but any preconception that this entails dilution of the mathematics (including number theory) is dispelled by the clear, complete, and efficient presentation.

There is an immediate definition and treatment of rings. The ring ℤ[i] of Gaussian integers a + bi “play[s] a fundamental role [typo corrected] in this book.” And “our bedrock” is the well-ordering principle.

The lemma (-a)(-b) = ab is proved for any commutative ring. Why two negatives yield a positive is an occasional question at parties. Campbell’s is the right approach to rendering hand-waving obsolete.

The exercises figure very large in this book, both by weight and by volume. Some results are substantive and interesting, such as Binet’s expression for the n’th Fibonacci number.

The buildup to, and proof of, the fundamental theorem of arithmetic (that is, the uniqueness of prime decomposition) is modern in the best sense of the word. The supplementary casting of the Euclidean algorithm in terms of continued fractions serves to enrich the student’s conceptual grasp and technical repertory. The “amazing array’s” use in unraveling continued fractions to yield rational numbers is a powerful example. The “amazing super-array” is a (well, amazing) generalization that would thrill any mathematician, computer scientist, or fellow traveler to work through.

In chapter 2, “Modular Arithmetic in ℤ/mℤ,” the treatment of modular arithmetic and congruences is again very rich. The many exercises achieve maximal pedagogy; the author’s request for readers’ conjectures ensures active reading.

Sections 23 and 24 treat solution of polynomial equations and systems of linear equations in ℤ/mℤ. What may be termed landmark theorems, such as the Chinese remainder theorem, (extension of) Wilson’s theorem, and Euler’s and Fermat’s (little) theorem, are treated clearly and efficiently.

Section 30, on divisibility, is a delight to follow page by page (with pencil).

Chapter 3, “Quadratic Extensions of the Integers, ℤ[√d],” is “a zoom out” from the ring of integers to the Gaussian integers and to quadratic extensions defined by the square roots of non-square integers.

Divisibility, Euclid’s algorithm, and factorization are treated superbly, particularly for the Gaussian ring ℤ[i]. One is confident that, in addition to the book’s hands-on tutelage regarding methods of proof (sans excess formalism), one comes away being prepared to discover applications, if such is one’s inclination. I’m reminded of the use of quaternions (a purely theoretical discovery of the 1830s) in late 20th-century avionic navigation systems.

Chapter 4, “An Interlude of Analytic Number Theory”: though “most of this course” deals with algebraic number theory, the chapter invokes the continuum, that is, the real numbers ℝ, in its treatment of the statistical distribution of primes. The proof of the existence of arbitrarily large gaps in the primes is a gem that illustrates by contrast the necessity of a statistical approach to, in my words, the decidedly nonrandom set of natural numbers ℕ.

As a quibble, the manifestly informal statement of Theorem 24 that “there are more primes than perfect squares” may evoke a double take in view of Georg Cantor’s seminal discovery that there are, for example, exactly as many primes as there are squares of integers, namely aleph-zero. The author’s precise meaning is, however, clear from the next line: the sum of all reciprocals of primes diverges, whilst the sum of all reciprocals of squares converges, as is proved in calculus/analysis.

Chapter 5 realizes “one of the high points of any first course in number theory,” namely the statement and proof of the law of quadratic reciprocity. It is a clear but high-density chapter whose interested parties are potential or actual number theory specialists. The reader will not be deprived of masterly treatments of the Legendre and Jacobi symbols.

Chapter 6, “Further Topics,” will put the diligent reader on a number-theoretic professional track, with such projects as p-adic numbers (and a slew of others).

This outstanding book will enlighten readers in every field that uses mathematics. The unity of mathematics is certainly not just a slogan, but is palpable in this excellent work, which has my highest recommendation.

Reviewer:  George Hacken Review #: CR146438 (1905-0159)
1) Kolata, G. Scientist at work: Andrew Wiles; math whiz who battled 350-year-old problem. New York Times June 29, 1993, https://nyti.ms/2UZVXo9.
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