The most important revision in the second edition are the enlargment of the treatment of p-adic functions in Ch. 4 to include the Iwasawa logarithm and the p-adic gamma-function, the rearrangement and addition of some exercises, the inclusion of an extensive appendix of answers and hints to the exercises, and numerous corrections and clarifications. The first edition has become the standard introduction to the theory of p-adic numbers on the advanced undergraduate and beginning graduate level.
Rezensionen / Stimmen
From the reviews of the second edition:
"In the second edition of this text, Koblitz presents a wide-ranging introduction to the theory of p-adic numbers and functions. . there are some really nice exercises that allow the reader to explore the material. . And with the exercises, the book would make a good textbook for a graduate course, provided the students have a decent background in analysis and number theory." (Donald L. Vestal, The Mathematical Association of America, April, 2011)
Reihe
Auflage
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Für Beruf und Forschung
Graduate
Illustrationen
Maße
Höhe: 241 mm
Breite: 160 mm
Dicke: 15 mm
Gewicht
ISBN-13
978-0-387-96017-3 (9780387960173)
DOI
10.1007/978-1-4612-1112-9
Schweitzer Klassifikation
I p-adic numbers.- 1. Basic concepts.- 2. Metrics on the rational numbers.- Exercises.- 3. Review of building up the complex numbers.- 4. The field of p-adic numbers.- 5. Arithmetic in ?p.- Exercises.- II p-adic interpolation of the Riemann zeta-function.- 1. A formula for ?(2k).- 2. p-adic interpolation of the function f(s) = as.- Exercises.- 3. p-adic distributions.- Exercises.- 4. Bernoulli distributions.- 5. Measures and integration.- Exercises.- 6. The p-adic ?-function as a Mellin-Mazur transform.- 7. A brief survey (no proofs).- Exercises.- III Building up ?.- 1. Finite fields.- Exercises.- 2. Extension of norms.- Exercises.- 3. The algebraic closure of ?p.- 4. ?.- Exercises.- IV p-adic power series.- 1. Elementary functions.- Exercises.- 2. The logarithm, gamma and Artin-Hasse exponential functions.- Exercises.- 3. Newton polygons for polynomials.- 4. Newton polygons for power series.- Exercises.- V Rationality of the zeta-function of a set of equations over a finite field.- 1. Hypersurfaces and their zeta-functions.- Exercises.- 2. Characters and their lifting.- 3. A linear map on the vector space of power series.- 4. p-adic analytic expression for the zeta-function.- Exercises.- 5. The end of the proof.- Answers and Hints for the Exercises.
