P-adic Numbers Are Of Great Theoretical Importance In Number Theory, Since They Allow The Use Of The Language Of Analysis To Study Problems Relating Toprime Numbers And Diophantine Equations. Further, They Offer A Realm Where One Can Do Things That Are Very Similar To Classical Analysis, But With Results That Are Quite Unusual. The Book Should Be Of Use To Students Interested In Number Theory, But At The Same Time Offers An Interesting Example Of The Many Connections Between Different Parts Of Mathematics. The Book Strives To Be Understandable To An Undergraduate Audience. Very Little Background Has Been Assumed, And The Presentation Is Leisurely. There Are Many Problems, Which Should Help Readers Who Are Working On Their Own (a Large Appendix With Hints On The Problem Is Included). Most Of All, The Book Should Offer Undergraduates Exposure To Some Interesting Mathematics Which Is Off The Beaten Track. Those Who Will Later Specialize In Number Theory, Algebraic Geometry, And Related Subjects Will Benefit More Directly, But All Mathematics Students Can Enjoy The Book. 1. Aperitif. 1.1. Hensel's Analogy. 1.2. Solving Congruences Modulo P[superscript N]. 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 Q[subscript P]. 3.4. Hensel's Lemma. 3.5. Local And Global -- 4. Elementary Analysis In Q[subscript P]. 4.1. Sequences And Series. 4.2. Power Series. 4.3. Some Elementary Functions. 4.4. 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 C[subscript P] -- 6. Analysis In C[subscript P]. 6.1. Almost Everything Extends. 6.2. Derivatives. 6.3. Deeper Results On Polynomials And Power Series. 6.4. Entire Functions. 6.5. Newton Polygons. 6.6. 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. Fernando Q. Gouvêa. Includes Bibliographical References (p. [277]-278) And Index. Front Matter....Pages i-vi Introduction....Pages 1-4 Apéritif....Pages 5-19 Foundations....Pages 21-39 p -adic Numbers....Pages 41-83 Elementary Analysis in Q p ....Pages 85-116 Vector Spaces and Field Extensions....Pages 117-169 Analysis in C p ....Pages 171-219 Back Matter....Pages 221-284 An elementary introduction to p-adic numbers, which are of great theoretical importance in number theory as they allow the study of problems relating to prime numbers and diophantine equations.
