內容簡介
高等數學中有各種數,有理數、實數、復數和p進數,其中p進數不被人所知,但其在數論和數學其它領域廣泛應用。本書是介紹p進數理論的入門性教科書,與其它同類書相比,本書內容較為淺顯易懂,適閤那些無意成為專傢但想對p進數有所瞭解的讀者。
作者簡介
本書作者Fernando Q. Gouvêa是數論和算術幾何領域的專傢,尤其緻力於研究模形式和伽羅瓦錶示,也涉獵數學史的研究工作。
目錄
1.1 Hensel's 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
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
5.6 Example: Adjoining a p—th R,oot 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
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