作者：（美）F.Q.戈维亚
页数：302
出版社：世界图书出版公司
出版日期：2017
ISBN：9787519226121
电子书格式：pdf/epub/txt
内容简介
高等数学中有各种数，有理数、实数、复数和p进数，其中p进数不被人所知，但其在数论和数学其它领域广泛应用。本书是介绍p进数理论的入门性教科书，与其它同类书相比，本书内容较为浅显易懂，适合那些无意成为专家但想对p进数有所了解的读者。
作者简介
本书作者Fernando Q. Gouvêa是数论和算术几何领域的专家，尤其致力于研究模形式和伽罗瓦表示，也涉猎数学史的研究工作。
本书特色
高等数学中有各种数，有理数、实数、复数和p进数，其中p进数最不被人所知，但其在数论和数学其它领域广泛应用。本书是介绍p进数理论的入门性教科书，与其它同类书相比，本书内容较为浅显易懂，适合那些无意成为专家但想对p进数有所了解的读者。
目录
Introduction
1 Aperitif
1.1 Hensers Analogy
1.2 Solving Congruences Modulo pn
1.3 Other Examples
2 Foundations
2.1 Absolute Values on a Field
2.2 Basic Properties
2.3 Topology
2.4 Algebra
3 p-adic Numbers
3.1 Absolute Values on Q
3.2 Completions
3.3 Exploring Qp
3.4 Hensel’s Lemma
3.5 Local and Global
4 Elementary Analysis in Qp
4.1 Sequences and Series
4.2 Functions, Continuity, Derivatives
4.3 Power Series
4.4 Functions Defined by Power Series
4.5 Some Elementary Functions
4.6 Interpolation
5 Vector Spaces and Field Extensions
5.1 Normed Vector Spaces over Complete Valued Fields
5.2 Finite-dimensional Normed Vector Spaces
5.3 Finite Field Extensions
5.4 Properties of Finite Extensions
5.5 Analysis
5.6 Example: Adjoining a p-th Root of Unity
5.7 On to Cp
6 Analysis in Cp
6.1 Almost Everything Extends
6.2 Deeper Results on Polynomials and Power Series
6.3 Entire Functions
6.4 Newton Polygons
6.5 Problems
A Hints and Comments on the Problems
B A Brief Glance at the Literature
B.1 Texts
B.2 Software
B.3 Other Books
Bibliography
Index
1 Aperitif
1.1 Hensers Analogy
1.2 Solving Congruences Modulo pn
1.3 Other Examples
2 Foundations
2.1 Absolute Values on a Field
2.2 Basic Properties
2.3 Topology
2.4 Algebra
3 p-adic Numbers
3.1 Absolute Values on Q
3.2 Completions
3.3 Exploring Qp
3.4 Hensel’s Lemma
3.5 Local and Global
4 Elementary Analysis in Qp
4.1 Sequences and Series
4.2 Functions, Continuity, Derivatives
4.3 Power Series
4.4 Functions Defined by Power Series
4.5 Some Elementary Functions
4.6 Interpolation
5 Vector Spaces and Field Extensions
5.1 Normed Vector Spaces over Complete Valued Fields
5.2 Finite-dimensional Normed Vector Spaces
5.3 Finite Field Extensions
5.4 Properties of Finite Extensions
5.5 Analysis
5.6 Example: Adjoining a p-th Root of Unity
5.7 On to Cp
6 Analysis in Cp
6.1 Almost Everything Extends
6.2 Deeper Results on Polynomials and Power Series
6.3 Entire Functions
6.4 Newton Polygons
6.5 Problems
A Hints and Comments on the Problems
B A Brief Glance at the Literature
B.1 Texts
B.2 Software
B.3 Other Books
Bibliography
Index
