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Algebraic Number Theory and Fermat's Last Theorem 4th edition [Kõva köide]

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(University of Warwick, UK), (University of Warwick, UK)
  • Formaat: Hardback, 342 pages, kõrgus x laius: 229x152 mm, kaal: 790 g, 4 Tables, black and white; 21 Illustrations, black and white
  • Ilmumisaeg: 13-Oct-2015
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1498738397
  • ISBN-13: 9781498738392
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Algebraic Number Theory and Fermat's Last Theorem 4th editionSuurem pilt
  • Formaat: Hardback, 342 pages, kõrgus x laius: 229x152 mm, kaal: 790 g, 4 Tables, black and white; 21 Illustrations, black and white
  • Ilmumisaeg: 13-Oct-2015
  • Kirjastus: Chapman & Hall/CRC
  • ISBN-10: 1498738397
  • ISBN-13: 9781498738392
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781498738392.html
Märksõnad:
Updated to reflect current research, Algebraic Number Theory and Fermats Last Theorem, Fourth Edition introduces fundamental ideas of algebraic numbers and explores one of the most intriguing stories in the history of mathematicsthe quest for a proof of Fermats Last Theorem. The authors use this celebrated theorem to motivate a general study of the theory of algebraic numbers from a relatively concrete point of view. Students will see how Wiless proof of Fermats Last Theorem opened many new areas for future work.

New to the Fourth Edition





Provides up-to-date information on unique prime factorization for real quadratic number fields, especially Harpers proof that Z(14) is Euclidean Presents an important new result: Mihilescus proof of the Catalan conjecture of 1844 Revises and expands one chapter into two, covering classical ideas about modular functions and highlighting the new ideas of Frey, Wiles, and others that led to the long-sought proof of Fermats Last Theorem Improves and updates the index, figures, bibliography, further reading list, and historical remarks

Written by preeminent mathematicians Ian Stewart and David Tall, this text continues to teach students how to extend properties of natural numbers to more general number structures, including algebraic number fields and their rings of algebraic integers. It also explains how basic notions from the theory of algebraic numbers can be used to solve problems in number theory.

Arvustused

"It is the discussion of [ Fermats Last Theorem], I think, that sets this book apart from others there are a number of other texts that introduce algebraic number theory, but I dont know of any others that combine that material with the kind of detailed exposition of FLT that is found here...To summarize and conclude: this is an interesting and attractive book. It would make an attractive text for an early graduate course on algebraic number theory, as well as a nice source of information for people interested in FLT, and especially its connections with algebraic numbers." Dr. Mark Hunacek, MAA Reviews, June 2016

Praise for Previous Editions"The book remains, as before, an extremely attractive introduction to algebraic number theory from the ideal-theoretic perspective." Andrew Bremner, Mathematical Reviews, February 2003 "It is the discussion of [ Fermats Last Theorem], I think, that sets this book apart from others there are a number of other texts that introduce algebraic number theory, but I dont know of any others that combine that material with the kind of detailed exposition of FLT that is found here...To summarize and conclude: this is an interesting and attractive book. It would make an attractive text for an early graduate course on algebraic number theory, as well as a nice source of information for people interested in FLT, and especially its connections with algebraic numbers." Dr. Mark Hunacek, MAA Reviews, June 2016

Praise for Previous Editions"The book remains, as before, an extremely attractive introduction to algebraic number theory from the ideal-theoretic perspective." Andrew Bremner, Mathematical Reviews, February 2003

Preface to the Third Edition ix
Preface to the Fourth Edition xv
Index of Notation xvii
The Origins of Algebraic Number Theory 1(8)
I Algebraic Methods
9(120)
1 Algebraic Background
11(26)
1.1 Rings and Fields
12(3)
1.2 Factorization of Polynomials
15(7)
1.3 Field Extensions
22(2)
1.4 Symmetric Polynomials
24(2)
1.5 Modules
26(2)
1.6 Free Abelian Groups
28(5)
1.7 Exercises
33(4)
2 Algebraic Numbers
37(26)
2.1 Algebraic Numbers
38(2)
2.2 Conjugates and Discriminants
40(3)
2.3 Algebraic Integers
43(4)
2.4 Integral Bases
47(3)
2.5 Norms and Traces
50(3)
2.6 Rings of Integers
53(6)
2.7 Exercises
59(4)
3 Quadratic and Cyclotomic Fields
63(12)
3.1 Quadratic Fields
63(3)
3.2 Cyclotomic Fields
66(5)
3.3 Exercises
71(4)
4 Factorization into Irreducibles
75(28)
4.1 Historical Background
77(1)
4.2 Trivial Factorizations
78(3)
4.3 Factorization into Irreducibles
81(3)
4.4 Examples of Non-Unique Factorization into Irreducibles
84(4)
4.5 Prime Factorization
88(4)
4.6 Euclidean Domains
92(1)
4.7 Euclidean Quadratic Fields
93(3)
4.8 Consequences of Unique Factorization
96(2)
4.9 The Ramanujan-Nagell Theorem
98(2)
4.10 Exercises
100(3)
5 Ideals
103(26)
5.1 Historical Background
104(3)
5.2 Prime Factorization of Ideals
107(9)
5.3 The Norm of an Ideal
116(8)
5.4 Non-Unique Factorization in Cyclotomic Fields
124(2)
5.5 Exercises
126(3)
II Geometric Methods
129(38)
6 Lattices
131(8)
6.1 Lattices
131(3)
6.2 The Quotient Torus
134(4)
6.3 Exercises
138(1)
7 Minkowski's Theorem
139(6)
7.1 Minkowski's Theorem
139(3)
7.2 The Two-Squares Theorem
142(1)
7.3 The Four-Squares Theorem
143(1)
7.4 Exercises
144(1)
8 Geometric Representation of Algebraic Numbers
145(6)
8.1 The Space Lst
145(5)
8.2 Exercises
150(1)
9 Class-Group and Class-Number
151(16)
9.1 The Class-Group
152(1)
9.2 An Existence Theorem
153(4)
9.3 Finiteness of the Class-Group
157(1)
9.4 How to Make an Ideal Principal
158(4)
9.5 Unique Factorization of Elements in an Extension Ring
162(2)
9.6 Exercises
164(3)
III Number-Theoretic Applications
167(110)
10 Computational Methods
169(14)
10.1 Factorization of a Rational Prime
169(3)
10.2 Minkowski Constants
172(4)
10.3 Some Class-Number Calculations
176(3)
10.4 Table of Class-Numbers
179(1)
10.5 Exercises
180(3)
11 Kummer's Special Case of Fermat's Last Theorem
183(18)
11.1 Some History
183(3)
11.2 Elementary Considerations
186(2)
11.3 Kummer's Lemma
188(5)
11.4 Kummer's Theorem
193(3)
11.5 Regular Primes
196(1)
11.6 Exercises
197(4)
12 The Path to the Final Breakthrough
201(14)
12.1 The Wolfskehl Prize
201(2)
12.2 Other Directions
203(2)
12.3 Modular Functions and Elliptic Curves
205(1)
12.4 The Taniyama-Shimura-Weil Conjecture
206(1)
12.5 Prey's Elliptic Equation
207(1)
12.6 The Amateur who Became a Model Professional
208(3)
12.7 Technical Hitch
211(1)
12.8 Flash of Inspiration
211(2)
12.9 Exercises
213(2)
13 Elliptic Curves
215(20)
13.1 Review of Conies
216(1)
13.2 Projective Space
217(5)
13.3 Rational Conics and the Pythagorean Equation
222(2)
13.4 Elliptic Curves
224(3)
13.5 The Tangent/Secant Process
227(1)
13.6 Group Structure on an Elliptic Curve
228(4)
13.7 Applications to Diophantine Equations
232(2)
13.8 Exercises
234(1)
14 Elliptic Functions
235(24)
14.1 Trigonometry Meets Diophantus
235(8)
14.2 Elliptic Functions
243(6)
14.3 Legendre and Weierstrass
249(2)
14.4 Modular Functions
251(5)
14.5 Exercises
256(3)
15 Wiles's Strategy and Recent Developments
259(18)
15.1 The Frey Elliptic Curve
259(2)
15.2 The Taniyama-Shimura-Weil Conjecture
261(3)
15.3 Sketch Proof of Fermat's Last Theorem
264(2)
15.4 Recent Developments
266(10)
15.5 Exercises
276(1)
IV Appendices
277(2)
A Quadratic Residues
279(20)
A.1 Quadratic Equations in Zm
280(2)
A.2 The Units of Zm
282(5)
A.3 Quadratic Residues
287(9)
A.4 Exercises
296(3)
B Dirichlet's Units Theorem
299(10)
B.1 Introduction
299(1)
B.2 Logarithmic Space
300(1)
B.3 Embedding the Unit Group in Logarithmic Space
301(1)
B.4 Dirichlet's Theorem
302(5)
B.5 Exercises
307(2)
Bibliography 309(8)
Index 317
Ian Stewart is an emeritus professor of mathematics at the University of Warwick and a fellow of the Royal Society. Dr. Stewart has been a recipient of many honors, including the Royal Societys Faraday Medal, the IMA Gold Medal, the AAAS Public Understanding of Science and Technology Award, and the LMS/IMA Zeeman Medal. He has published more than 180 scientific papers and numerous books, including several bestsellers co-authored with Terry Pratchett and Jack Cohen that combine fantasy with nonfiction.

David Tall is an emeritus professor of mathematical thinking at the University of Warwick. Dr. Tall has published numerous mathematics textbooks and more than 200 papers on mathematics and mathematics education. His research interests include cognitive theory, algebra, visualization, mathematical thinking, and mathematics education.