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E-raamat: Number Theory: An Introduction via the Density of Primes

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  • Formaat: PDF+DRM
  • Ilmumisaeg: 19-Sep-2016
  • Kirjastus: Birkhauser Verlag AG
  • Keel: eng
  • ISBN-13: 9783319438757
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Number Theory: An Introduction via the Density of PrimesSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 19-Sep-2016
  • Kirjastus: Birkhauser Verlag AG
  • Keel: eng
  • ISBN-13: 9783319438757
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Now in its second edition, this textbook provides an introduction and overview of number theory based on the density and properties of the prime numbers. This unique approach offers both a firm background in the standard material of number theory, as well as an overview of the entire discipline. All of the essential topics are covered, such as the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. New in this edition are coverage of p-adic numbers, Hensel's lemma, multiple zeta-values, and elliptic curve methods in primality testing.

Key topics and features include:
  • A solid introduction to analytic number theory, including full proofs of Dirichlet's Theorem and the Prime Number Theorem
  • Concise treatment of algebraic number theory, including a complete presentation of primes, prime factorizations in algebraic number fields, and unique factorization of ideals
  • Discussion of the AKS algorithm, which shows that primality testing is one of polynomial time, a topic not usually included in such texts
  • Many interesting ancillary topics, such as primality testing and cryptography, Fermat and Mersenne numbers, and Carmichael numbers
The user-friendly style, historical context, and wide range of exercises that range from simple to quite difficult (with solutions and hints provided for select exercises) make Number Theory: An Introduction via the Density of Primes ideal for both self-study and classroom use. Intended for upper level undergraduates and beginning graduates, the only prerequisites are a basic knowledge of calculus, multivariable calculus, and some linear algebra. All necessary concepts from abstract algebra and complex analysis are introduced where needed.

Arvustused

In this text, Fine (mathematics, Fairfield Univ.) and Rosenberger (Univ. of Hamburg, Germany) successfully present number theory from the inception of primes to recent developments in algebraic and analytic number theory and cryptography. Numerous exercises and open problems are provided. The breadth and depth of topics covered are impressive, making this an excellent text for those interested in the field of number theory. Summing Up: Recommended. Upper-division undergraduates and graduate students. (J. T. Zerger, Choice, Vol. 54 (9), May, 2017)

The book is chatty and leisurely, with lots of historical notes and lots of worked examples. The exercises at the end of each chapter are good and there are a reasonable number of them. a good text for an introductory course . (Allen Stenger, MAA Reviews, maa.org, November, 2016)

1 Introduction and Historical Remarks
1(6)
2 Basic Number Theory
7(52)
2.1 The Ring of Integers
7(3)
2.2 Divisibility, Primes, and Composites
10(6)
2.3 The Fundamental Theorem of Arithmetic
16(6)
2.4 Congruences and Modular Arithmetic
22(17)
2.4.1 Basic Theory of Congruences
22(1)
2.4.2 The Ring of Integers Mod N
23(4)
2.4.3 Units and the Euler Phi Function
27(5)
2.4.4 Fermat's Little Theorem and the Order of an Element
32(4)
2.4.5 On Cyclic Groups
36(3)
2.5 The Solution of Polynomial Congruences Modulo m
39(9)
2.5.1 Linear Congruences and the Chinese Remainder Theorem
39(6)
2.5.2 Higher Degree Congruences
45(3)
2.6 Quadratic Reciprocity
48(7)
2.7 Exercises
55(4)
3 The Infinitude of Primes
59(84)
3.1 The Infinitude of Primes
59(33)
3.1.1 Some Direct Proofs and Variations
59(3)
3.1.2 Some Analytic Proofs and Variations
62(4)
3.1.3 The Fermat and Mersenne Numbers
66(5)
3.1.4 The Fibonacci Numbers and the Golden Section
71(13)
3.1.5 Some Simple Cases of Dirichlet's Theorem
84(5)
3.1.6 A Topological Proof and a Proof Using Codes
89(3)
3.2 Sums of Squares
92(20)
3.2.1 Pythagorean Triples
93(3)
3.2.2 Fermat's Two-Square Theorem
96(4)
3.2.3 The Modular Group
100(7)
3.2.4 Lagrange's Four Square Theorem
107(3)
3.2.5 The Infinitude of Primes Through Continued Fractions
110(2)
3.3 Dirichlet's Theorem
112(19)
3.4 Twin Prime Conjecture and Related Ideas
131(1)
3.5 Primes Between x and 2x
132(1)
3.6 Arithmetic Functions and the Mobius Inversion Formula
133(5)
3.7 Exercises
138(5)
4 The Density of Primes
143(76)
4.1 The Prime Number Theorem---Estimates and History
143(4)
4.2 Chebyshev's Estimate and Some Consequences
147(12)
4.3 Equivalent Formulations of the Prime Number Theorem
159(10)
4.4 The Riemann Zeta Function and the Riemann Hypothesis
169(17)
4.4.1 The Real Zeta Function of Euler
170(5)
4.4.2 Analytic Functions and Analytic Continuation
175(4)
4.4.3 The Riemann Zeta Function
179(7)
4.5 The Prime Number Theorem
186(7)
4.6 The Elementary Proof
193(5)
4.7 Multiple Zeta Values
198(8)
4.8 Some Extensions and Comments
206(7)
4.9 Exercises
213(6)
5 Primality Testing---An Overview
219(66)
5.1 Primality Testing and Factorization
219(1)
5.2 Sieving Methods
220(16)
5.2.1 Brun's Sieve and Brun's Theorem
226(10)
5.3 Primality Testing and Prime Records
236(27)
5.3.1 Pseudo-Primes and Probabilistic Testing
241(8)
5.3.2 The Lucas--Lehmer Test and Prime Records
249(6)
5.3.3 Some Additional Primality Tests
255(2)
5.3.4 Elliptic Curve Methods
257(6)
5.4 Cryptography and Primes
263(7)
5.4.1 Some Number Theoretic Cryptosystems
267(3)
5.5 Public Key Cryptography and the RSA Algorithm
270(3)
5.6 Elliptic Curve Cryptography
273(3)
5.7 The AKS Algorithm
276(6)
5.8 Exercises
282(3)
6 Primes and Algebraic Number Theory
285(86)
6.1 Algebraic Number Theory
285(2)
6.2 Unique Factorization Domains
287(21)
6.2.1 Euclidean Domains and the Gaussian Integers
293(8)
6.2.2 Principal Ideal Domains
301(3)
6.2.3 Prime and Maximal Ideals
304(4)
6.3 Algebraic Number Fields
308(21)
6.3.1 Algebraic Extensions of Q
316(3)
6.3.2 Algebraic and Transcendental Numbers
319(2)
6.3.3 Symmetric Polynomials
321(4)
6.3.4 Discriminant and Norm
325(4)
6.4 Algebraic Integers
329(19)
6.4.1 The Ring of Algebraic Integers
331(2)
6.4.2 Integral Bases
333(2)
6.4.3 Quadratic Fields and Quadratic Integers
335(4)
6.4.4 The Transcendence of e and π
339(3)
6.4.5 The Geometry of Numbers---Minkowski Theory
342(3)
6.4.6 Dirichlet's Unit Theorem
345(3)
6.5 The Theory of Ideals
348(18)
6.5.1 Unique Factorization of Ideals
350(7)
6.5.2 An Application of Unique Factorization
357(2)
6.5.3 The Ideal Class Group
359(2)
6.5.4 Norms of Ideals
361(3)
6.5.5 Class Number
364(2)
6.6 Exercises
366(5)
7 The Fields Qp of p-Adic Numbers: Hensel's Lemma
371(38)
7.1 The p-Adic Fields and p-Adic Expansions
371(2)
7.2 The Construction of the Real Numbers
373(8)
7.2.1 The Completeness of Real Numbers
373(3)
7.2.2 The Construction of R
376(5)
7.2.3 The Characterization of R
381(1)
7.3 Normed Fields and Cauchy Completions
381(1)
7.4 The p-Adic Fields
382(5)
7.4.1 The p-Adic Norm
385(2)
7.5 The Construction of Qp
387(7)
7.5.1 p-Adic Arithmetic and p-Adic Expansions
387(7)
7.6 The p-Adic Integers
394(4)
7.6.1 Principal Ideals and Unique Factorization
396(1)
7.6.2 The Completeness of Zp
397(1)
7.7 Ostrowski's Theorem
398(1)
7.8 Hensel's Lemma and Applications
398(5)
7.8.1 The Non-isomorphism of the p-Adic Fields
402(1)
7.9 Exercises
405(4)
Index 409
Benjamin Fine, PhD, is Professor of Mathematics at Fairfield University, CT, USA.

Gerhard Rosenberger, PhD, is Professor (retired) at Dortmund University of Technology, Germany.
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