Muutke küpsiste eelistusi

Number Theory [Pehme köide]

,
  • Formaat: Paperback / softback, 552 pages, kaal: 983 g
  • Sari: Pure and Applied Undergraduate Texts
  • Ilmumisaeg: 30-Jan-2021
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470452758
  • ISBN-13: 9781470452759
Teised raamatud teemal:
Number TheorySuurem pilt
  • Formaat: Paperback / softback, 552 pages, kaal: 983 g
  • Sari: Pure and Applied Undergraduate Texts
  • Ilmumisaeg: 30-Jan-2021
  • Kirjastus: American Mathematical Society
  • ISBN-10: 1470452758
  • ISBN-13: 9781470452759
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781470452759.html
Märksõnad:
Number Theory is a newly translated and revised edition of the most popular introductory textbook on the subject in Hungary. The book covers the usual topics of introductory number theory: divisibility, primes, Diophantine equations, arithmetic functions, and so on. It also introduces several more advanced topics including congruences of higher degree, algebraic number theory, combinatorial number theory, primality testing, and cryptography. The development is carefully laid out with ample illustrative examples and a treasure trove of beautiful and challenging problems. The exposition is both clear and precise.

The book is suitable for both graduate and undergraduate courses with enough material to fill two or more semesters and could be used as a source for independent study and capstone projects. Freud and Gyarmati are well-known mathematicians and mathematical educators in Hungary, and the Hungarian version of this book is legendary there. The authors personal pedagogical style as a facet of the rich Hungarian tradition shines clearly through. It will inspire and exhilarate readers.

Arvustused

I think the book is not only the best book on number theory, but the best textbook I have ever seen. Beginning students can gain a solid foundation in number theory, advanced students can challenge themselves with the often deep and always delightful exercises, and everyone, including experts in the field, can discover new topics or attain a better understanding of familiar ones. As with masterpieces in music in literature, one gets more out of it with each additional visit." - Béla Bajnok, Gettysburg College, The American Mathematical Monthly

Introduction 1(1)
Structure of the book 1(1)
Exercises 2(1)
Short overview of the individual chapters 2(2)
Technical details 4(1)
Commemoration 4(1)
Acknowledgements 5(2)
Chapter 1 Basic Notions
7(30)
1.1 Divisibility
7(2)
Exercises 1.1
9(2)
1.2 Division Algorithm
11(2)
Exercises 1.2
13(2)
1.3 Greatest Common Divisor
15(4)
Exercises 1.3
19(2)
1.4 Irreducible and Prime Numbers
21(2)
Exercises 1.4
23(1)
1.5 The Fundamental Theorem of Arithmetic
24(3)
Exercises 1.5
27(1)
1.6 Standard Form
28(5)
Exercises 1.6
33(4)
Chapter 2 Congruences
37(36)
2.1 Elementary Properties
37(3)
Exercises 2.1
40(1)
2.2 Residue Systems and Residue Classes
41(3)
Exercises 2.2
44(2)
2.3 Euler's Function φ
46(3)
Exercises 2.3
49(1)
2.4 The Euler--Fermat Theorem
50(1)
Exercises 2.4
51(1)
2.5 Linear Congruences
52(5)
Exercises 2.5
57(1)
2.6 Simultaneous Systems of Congruences
58(6)
Exercises 2.6
64(2)
2.7 Wilson's Theorem
66(1)
Exercises 2.7
67(1)
2.8 Operations with Residue Classes
68(2)
Exercises 2.8
70(3)
Chapter 3 Congruences of Higher Degree
73(28)
3.1 Number of Solutions and Reduction
73(2)
Exercises 3.1
75(1)
3.2 Order
76(2)
Exercises 3.2
78(2)
3.3 Primitive Roots
80(4)
Exercises 3.3
84(2)
3.4 Discrete Logarithm (Index)
86(1)
Exercises 3.4
87(1)
3.5 Binomial Congruences
88(2)
Exercises 3.5
90(1)
3.6 Chevalley's Theorem, Konig--Rados Theorem
91(4)
Exercises 3.6
95(1)
3.7 Congruences with Prime Power Moduli
96(2)
Exercises 3.7
98(3)
Chapter 4 Legendre and Jacobi Symbols
101(12)
4.1 Quadratic Congruences
101(2)
Exercises 4.1
103(1)
4.2 Quadratic Reciprocity
104(4)
Exercises 4.2
108(1)
4.3 Jacobi Symbol
109(2)
Exercises 4.3
111(2)
Chapter 5 Prime Numbers
113(52)
5.1 Classical Problems
113(4)
Exercises 5.1
117(1)
5.2 Fermat and Mersenne Primes
118(6)
Exercises 5.2
124(1)
5.3 Primes in Arithmetic Progressions
125(2)
Exercises 5.3
127(1)
5.4 How Big Is π(x)?
128(5)
Exercises 5.4
133(1)
5.5 Gaps between Consecutive Primes
134(5)
Exercises 5.5
139(1)
5.6 The Sum of Reciprocals of Primes
140(7)
Exercises 5.6
147(2)
5.7 Primality Tests
149(8)
Exercises 5.7
157(3)
5.8 Cryptography
160(3)
Exercises 5.8
163(2)
Chapter 6 Arithmetic Functions
165(46)
6.1 Multiplicative and Additive Functions
165(2)
Exercises 6.1
167(3)
6.2 Some Important Functions
170(3)
Exercises 6.2
173(2)
6.3 Perfect Numbers
175(2)
Exercises 6.3
177(1)
6.4 Behavior of d(n)
178(7)
Exercises 6.4
185(1)
6.5 Summation and Inversion Functions
186(3)
Exercises 6.5
189(1)
6.6 Convolution
190(3)
Exercises 6.6
193(2)
6.7 Mean Value
195(11)
Exercises 6.7
206(1)
6.8 Characterization of Additive Functions
207(2)
Exercises 6.8
209(2)
Chapter 7 Diophantine Equations
211(52)
7.1 Linear Diophantine Equation
212(2)
Exercises 7.1
214(1)
7.2 Pythagorean Triples
215(2)
Exercises 7.2
217(1)
7.3 Some Elementary Methods
218(3)
Exercises 7.3
221(2)
7.4 Gaussian Integers
223(6)
Exercises 7.4
229(1)
7.5 Sums of Squares
230(5)
Exercises 7.5
235(1)
7.6 Waring's Problem
236(4)
Exercises 7.6
240(1)
7.7 Fermat's Last Theorem
241(8)
Exercises 7.7
249(2)
7.8 Pell's Equation
251(4)
Exercises 7.8
255(1)
7.9 Partitions
256(5)
Exercises 7.9
261(2)
Chapter 8 Diophantine Approximation
263(22)
8.1 Approximation of Irrational Numbers
263(5)
Exercises 8.1
268(2)
8.2 Minkowski's Theorem
270(4)
Exercises 8.2
274(1)
8.3 Continued Fractions
275(5)
Exercises 8.3
280(1)
8.4 Distribution of Fractional Parts
281(2)
Exercises 8.4
283(2)
Chapter 9 Algebraic and Transcendental Numbers
285(26)
9.1 Algebraic Numbers
285(3)
Exercises 9.1
288(1)
9.2 Minimal Polynomial and Degree
288(2)
Exercises 9.2
290(1)
9.3 Operations with Algebraic Numbers
291(3)
Exercises 9.3
294(2)
9.4 Approximation of Algebraic Numbers
296(4)
Exercises 9.4
300(1)
9.5 Transcendence of e
301(5)
Exercises 9.5
306(1)
9.6 Algebraic Integers
306(2)
Exercises 9.6
308(3)
Chapter 10 Algebraic Number Fields
311(30)
10.1 Field Extensions
311(3)
Exercises 10.1
314(1)
10.2 Simple Algebraic Extensions
315(4)
Exercises 10.2
319(1)
10.3 Quadratic Fields
320(10)
Exercises 10.3
330(1)
10.4 Norm
331(3)
Exercises 10.4
334(1)
10.5 Integral Basis
335(5)
Exercises 10.5
340(1)
Chapter 11 Ideals
341(36)
11.1 Ideals and Factor Rings
341(4)
Exercises 11.1
345(2)
11.2 Elementary Connections to Number Theory
347(3)
Exercises 11.2
350(1)
11.3 Unique Factorization, Principal Ideal Domains, and Euclidean Rings
350(5)
Exercises 11.3
355(2)
11.4 Divisibility of Ideals
357(4)
Exercises 11.4
361(2)
11.5 Dedekind Rings
363(9)
Exercises 11.5
372(1)
11.6 Class Number
373(3)
Exercises 11.6
376(1)
Chapter 12 Combinatorial Number Theory
377(44)
12.1 All Sums Are Distinct
377(7)
Exercises 12.1
384(2)
12.2 Sidon Sets
386(7)
Exercises 12.2
393(1)
12.3 Sumsets
394(8)
Exercises 12.3
402(1)
12.4 Schur's Theorem
403(4)
Exercises 12.4
407(1)
12.5 Covering Congruences
408(4)
Exercises 12.5
412(1)
12.6 Additive Complements
412(6)
Exercises 12.6
418(3)
Answers and Hints 421(110)
A.1 Basic Notions
421(10)
A.2 Congruences
431(11)
A.3 Congruences of Higher Degree
442(10)
A.4 Legendre and Jacobi Symbols
452(3)
A.5 Prime Numbers
455(12)
A.6 Arithmetic Functions
467(16)
A.7 Diophantine Equations
483(18)
A.8 Diophantine Approximation
501(4)
A.9 Algebraic and Transcendental Numbers
505(5)
A.10 Algebraic Number Fields
510(6)
A.11 Ideals
516(5)
A.12 Combinatorial Number Theory
521(10)
Historical Notes 531(6)
Tables 537(1)
Primes 2--1733 538(1)
Primes 1741--3907 539(1)
Prime Factorization 540(1)
Mersenne Numbers 541(1)
Fermat Numbers 542(1)
Index 543
Robert Freud, University Eotvos Lorand, Budapest, Hungary and Edit Gyarmati