Algebraic number theory /

Undergraduate mathematics students need both to develop facility with numerical and symbolic calculation and comfort with abstraction. Algebraic number theory offers an ideal context for encountering the synthesis of these goals. One could compile a shelf of graduate-level expositions of algebraic number theory, and another shelf of undergraduate general number theory texts that culminate with a first exposure to it. Four features makes this the rare undergraduate-level introduction: a leisurely pace; the presence of all foundational material (both algebraic and number-theoretical) necessary for a self-contained treatment; the inclusion of, not just mere examples, but detailed case studies (which graduate texts would either reduce to exercises or omit entirely, deferring to complete treatments in even more advanced monographs); and an ample supply of worked exercises. Of the case studies, imaginary quadratic fields get the most attention, with additional sections on real quadratic, biquadratic, cubic, and cyclotomic fields. Final chapters introduce analytic methods and applications to integer factorization. Students will need Galois theory for graduate-level sequels, as Jarvis (Univ. of Sheffield, UK) sidesteps it by concentrating on low-degree fields, much as M. Trifkovic does in Algebraic Theory of Quadratic Numbers (CH, May'14, 51-5073). As they work different examples, these competing books complement one another. Summing Up: Highly recommended. Upper-division undergraduates. --David V. Feldman, University of New Hampshire