Fourier Analysis on Number Fields, by D. Ramakrishnan and R. J. Valenza
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 37, Number 3, Pages 373{377 S 0273-0979(00)00872-7 Article electronically published on April 7, 2000 Fourier analysis on number fields, by D. Ramakrishnan and R. J. Valenza, Springer, New York, 1999, xxi + 350 pp., $39.95, ISBN 0-387-98436-4 Dinakar Ramakrishnan and Robert J. Valenza's recent book Fourier Analysis on Number Fields ([RV]) is an introduction to number theory organized around John Tate's 1950 Princeton Ph.D. thesis [T]. Less comprehensive than Weil's famous book Basic Number Theory ([W]), Ramakrishnan and Valenza's book is notable for the thoroughness with which it treats the analytic background necessary to fully appreciate the technicalities of Tate's methods. Tate's thesis, \Fourier Analysis in Number Fields and Hecke's Zeta-Functions", combined techniques of abstract Fourier analysis with the valuation-theoretic ap- proach to algebraic number theory developed by Chevalley and Artin and Whaples to obtain an extremely powerful and elegant approach to the theory of zeta func- tions. The approach pioneered by Tate in his thesis has proved to be extremely in- fluential in many later developments in number theory. Ramakrishnan and Valenza convey an appreciation for the depth of the analysis underlying Tate's work and the power and clarity of Tate's methods. The reader who masters the material in Ramakrishnan and Valenza's text will have no trouble going on to read the more general results, including the proofs of the main theorems of Class Field Theory, as given in Weil's book.
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