P-adic analysis compared with real / Svetlana Katok.

By:

Katok, Svetlana [author]

Material type: Text

TextSeries: Student mathematical library ; 37.Publication details: Providence, R.I. ; New Delhi ; American Mathematical Society, 2019.Description: xiii, 152 pages : illustrations ; 22 cmISBN:

9780821852248

Subject(s):

p-adic analysis

DDC classification:

22 512.74 K156P

Contents:

Foreword : MASS and REU at Penn State University -- Preface -- ch. 1. Arithmetic of the p-adic numbers -- 1.1. From Q to R ; the concept of completion -- Exercises 1-8 -- 1.2. Normed fields -- Exercises 9-16 -- 1.3. Construction of the completion of a normed field -- Exercises 17-19 -- 1.4. The field of p-adic numbers Qp -- Exercises 20-25 -- 1.5. Arithmetical operations in Qp -- Exercises 26-31 -- 1.6. The p-adic expansion of rational numbers -- Exercises 32-34 -- 1.7. Hensel's Lemma and congruences -- Exercises 35-44 -- 1.8. Algebraic properties of p-adic integers -- 1.9. Metrics and norms on the rational numbers. Ostrowski's theorem -- Exercises 45-46 -- 1.10. A digression : what about Qg if g is not a prime? -- Exercises 47-50 -- ch. 2. The topology of Qp vs. the topology of R -- 2.1. Elementary topological properties -- Exercises 51-53 -- c.2. Cantor sets -- Exercises 54-65 -- 2.3. Euclidean models of Zp -- Exercises 66-68. ch. 3. Elementary analysis in Qp -- 3.1. Sequences and series -- Exercises 69-73 -- 3.2. p-adic power series -- Exercises 74-78 -- 3.3. Can a p-adic power series be analytically continued? -- 3.4. Some elementary functions -- Exercises 79-81 -- 3.5. Further properties of p-adic exponential and logarithm -- 3.6. Zeros of p-adic power series -- Exercises 82-83 -- ch. 4. p-adic functions -- 4.1. Locally constant functions -- Exercises 84-87 -- 4.2. Continuous and uniformly continuous functions -- Exercises 88-90 -- 4.3. Points of discontinuity and the Baire Category Theorem -- Exercises 91-96 -- 4.4. Differentiability of p-adic functions -- 4.5. Isometries of Qp -- Exercises 97-100 -- 4.6. Interpolation -- Exercises 101-103 -- Answers, hints, and solutions for selected exercises -- Bibliography -- Index.

Summary: "The book gives an introduction to p-adic numbers from the point of view of number theory, topology, and analysis. Compared to other books on the subject, its novelty is both a particularly balanced approach to these three points of view and an emphasis on topics accessible to undergraduates. In addition, several topics from real analysis and elementary topology which are not usually covered in undergraduate courses (totally disconnected spaces and Cantor sets, points of discontinuity of maps and the Baire Category Theorem, surjectivity isometries of compact metric spaces) are also included in the book. They will enhance the reader's understanding of real analysis and intertwine the real and p-adic contexts of the book." "The book includes a large number of exercises. Answers, hints, and solutions for most of them appear at the end of the book. The book can be successfully used in a topic course or for self-study."--Jacket.

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Holdings
Item type	Current library	Call number	Vol info	Status	Date due	Barcode
Gift Books	Library and Documentation Division PGRRL	512.74 K156P (Browse shelf(Opens below))	V.37	Available		G016887

Gifted by NBHM

Foreword : MASS and REU at Penn State University --
Preface --
ch. 1. Arithmetic of the p-adic numbers --
1.1. From Q to R ; the concept of completion --
Exercises 1-8 --
1.2. Normed fields --
Exercises 9-16 --
1.3. Construction of the completion of a normed field --
Exercises 17-19 --
1.4. The field of p-adic numbers Qp --
Exercises 20-25 --
1.5. Arithmetical operations in Qp --
Exercises 26-31 --
1.6. The p-adic expansion of rational numbers --
Exercises 32-34 --
1.7. Hensel's Lemma and congruences --
Exercises 35-44 --
1.8. Algebraic properties of p-adic integers --
1.9. Metrics and norms on the rational numbers. Ostrowski's theorem --
Exercises 45-46 --
1.10. A digression : what about Qg if g is not a prime? --
Exercises 47-50 --
ch. 2. The topology of Qp vs. the topology of R --
2.1. Elementary topological properties --
Exercises 51-53 --
c.2. Cantor sets --
Exercises 54-65 --
2.3. Euclidean models of Zp --
Exercises 66-68. ch. 3. Elementary analysis in Qp --
3.1. Sequences and series --
Exercises 69-73 --
3.2. p-adic power series --
Exercises 74-78 --
3.3. Can a p-adic power series be analytically continued? --
3.4. Some elementary functions --
Exercises 79-81 --
3.5. Further properties of p-adic exponential and logarithm --
3.6. Zeros of p-adic power series --
Exercises 82-83 --
ch. 4. p-adic functions --
4.1. Locally constant functions --
Exercises 84-87 --
4.2. Continuous and uniformly continuous functions --
Exercises 88-90 --
4.3. Points of discontinuity and the Baire Category Theorem --
Exercises 91-96 --
4.4. Differentiability of p-adic functions --
4.5. Isometries of Qp --
Exercises 97-100 --
4.6. Interpolation --
Exercises 101-103 --
Answers, hints, and solutions for selected exercises --
Bibliography --
Index.

"The book gives an introduction to p-adic numbers from the point of view of number theory, topology, and analysis. Compared to other books on the subject, its novelty is both a particularly balanced approach to these three points of view and an emphasis on topics accessible to undergraduates. In addition, several topics from real analysis and elementary topology which are not usually covered in undergraduate courses (totally disconnected spaces and Cantor sets, points of discontinuity of maps and the Baire Category Theorem, surjectivity isometries of compact metric spaces) are also included in the book. They will enhance the reader's understanding of real analysis and intertwine the real and p-adic contexts of the book." "The book includes a large number of exercises. Answers, hints, and solutions for most of them appear at the end of the book. The book can be successfully used in a topic course or for self-study."--Jacket.

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