Pub. Date:
Springer New York
p-adic Numbers, p-adic Analysis, and Zeta-Functions / Edition 2

p-adic Numbers, p-adic Analysis, and Zeta-Functions / Edition 2

by Neal Koblitz


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p-adic Numbers, p-adic Analysis, and Zeta-Functions / Edition 2

The first edition of this work has become the standard introduction to the theory of p-adic numbers at both the advanced undergraduate and beginning graduate level. This second edition includes a deeper 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, as well as numerous clarifications.

Product Details

ISBN-13: 9780387960173
Publisher: Springer New York
Publication date: 04/11/1996
Series: Graduate Texts in Mathematics , #58
Edition description: 2nd Corrected ed. 1984. Corr. 2nd printing 1996
Pages: 153
Sales rank: 396,526
Product dimensions: 6.40(w) x 9.30(h) x 0.70(d)

Table of Contents

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.

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