Number Theory

viiiinstance) A(X) = A d X d +A d−1 X d−1 +· · ·+A 0 , so that the ith elementarysymmetric function of the roots is equal to (−1) i A d−i /A d .AcknowledgmentsA large part of the material on local fields has been taken with little changefrom the remarkable book by Cassels [Cas1], and also from unpublished notesof Jaulent written in 1994. For p-adic analysis, I have also liberally borrowedfrom work of Robert, in particular his superb GTM volume [Rob1]. For partof the material on elliptic curves I have borrowed from another excellent bookby Cassels [Cas2], as well as the treatises of Cremona and Silverman [Cre2],[Sil1], [Sil2], and the introductory book by Silverman–Tate [Sil-Tat]. I havealso borrowed from the classical books by Borevich–Shafarevich [Bor-Sha],Serre [Ser1], Ireland–Rosen [Ire-Ros], and Washington [Was]. I would liketo thank my former students K. Belabas, C. Delaunay, S. Duquesne, andD. Simon, who have helped me to write specific sections, and my colleagues J.-F. Jaulent and J. Martinet for answering many questions in algebraic numbertheory. I would also like to thank M. Bennett, J. Cremona, A. Kraus, andF. Rodriguez-Villegas for valuable comments on parts of this book. I wantespecially like to thank D. Bernardi for his thorough rereading of the firstten chapters of the manuscript, which enabled me to remove a large numberof errors, mathematical or otherwise.It is unavoidable that there still remain errors, typographical or otherwise,and the author would like to hear about them. Please send e-mail toHenri.Cohen@math.u-bordeaux1.frLists of known errors for the author’s books including the present one canbe obtained on the author’s home page at the URLhttp://www.math.u-bordeaux1.fr/~cohen/