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Cover: 9781461270140 | p-adic Numbers, p-adic Analysis, and Zeta-Functions | Neal Koblitz
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p-adic Numbers, p-adic Analysis, and Zeta-Functions
Taschenbuch von Neal Koblitz
Sprache: Englisch

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Beschreibung
Neal Koblitz was a student of Nicholas M. Katz, under whom he received his Ph.D. in mathematics at Princeton in 1974. He spent the year 1974 -75 and the spring semester 1978 in Moscow, where he did research in p -adic analysis and also translated Yu. I. Manin's "Course in Mathematical Logic" (GTM 53). He taught at Harvard from 1975 to 1979, and since 1979 has been at the University of Washington in Seattle. He has published papers in number theory, algebraic geometry, and p-adic analysis, and he is the author of "p-adic Analysis: A Short Course on Recent Work" (Cambridge University Press and GTM 97: "Introduction to Elliptic Curves and Modular Forms (Springer-Verlag).
Neal Koblitz was a student of Nicholas M. Katz, under whom he received his Ph.D. in mathematics at Princeton in 1974. He spent the year 1974 -75 and the spring semester 1978 in Moscow, where he did research in p -adic analysis and also translated Yu. I. Manin's "Course in Mathematical Logic" (GTM 53). He taught at Harvard from 1975 to 1979, and since 1979 has been at the University of Washington in Seattle. He has published papers in number theory, algebraic geometry, and p-adic analysis, and he is the author of "p-adic Analysis: A Short Course on Recent Work" (Cambridge University Press and GTM 97: "Introduction to Elliptic Curves and Modular Forms (Springer-Verlag).
Zusammenfassung
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.
Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2012
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 168
Reihe: Graduate Texts in Mathematics
Inhalt: xii
153 S.
ISBN-13: 9781461270140
ISBN-10: 1461270146
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Koblitz, Neal
Auflage: 2nd ed. 1984
Hersteller: Springer US
Springer New York
Graduate Texts in Mathematics
Maße: 235 x 155 x 10 mm
Von/Mit: Neal Koblitz
Erscheinungsdatum: 04.10.2012
Gewicht: 0,265 kg
preigu-id: 106118726
Zusammenfassung
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.
Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2012
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 168
Reihe: Graduate Texts in Mathematics
Inhalt: xii
153 S.
ISBN-13: 9781461270140
ISBN-10: 1461270146
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Koblitz, Neal
Auflage: 2nd ed. 1984
Hersteller: Springer US
Springer New York
Graduate Texts in Mathematics
Maße: 235 x 155 x 10 mm
Von/Mit: Neal Koblitz
Erscheinungsdatum: 04.10.2012
Gewicht: 0,265 kg
preigu-id: 106118726
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