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E-raamat: Certain Number-Theoretic Episodes In Algebra, Second Edition

  • Formaat: 444 pages
  • Ilmumisaeg: 19-Mar-2019
  • Kirjastus: CRC Press
  • Keel: eng
  • ISBN-13: 9781351023320
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Certain Number-Theoretic Episodes In Algebra, Second EditionSuurem pilt
  • Formaat: 444 pages
  • Ilmumisaeg: 19-Mar-2019
  • Kirjastus: CRC Press
  • Keel: eng
  • ISBN-13: 9781351023320
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The book attempts to point out the interconnections between number theory and algebra with a view to making a student understand certain basic concepts in the two areas forming the subject-matter of the book.
Preface xv
Acknowledgment xxi
About the Author xxiii
Chapter-Wise Description of the Contents xxv
Section I ELEMENTS OF THE THEORY OF NUMBERS
1(160)
1 From Euclid to Lucas: Elementary Theorems Revisited
3(30)
Introduction
3(3)
1.1 The quotient ring Z/rZ (r < 1)
6(5)
1.2 Congruences modulo a prime
11(5)
1.3 Fermat's two-squares theorem
16(3)
1.4 Lagrange's four-squares theorem
19(3)
1.5 Worked-out examples
22(4)
1.6 Notes / Remarks
26(7)
Exercises
29(1)
References
30(3)
2 Solutions of Congruences, Primitive Roots
33(8)
Introduction
33(1)
2.1 Theorems on congruences
34(3)
2.2 Worked-out examples
37(1)
2.3 Notes / Remarks
38(3)
Exercises
39(1)
References
40(1)
3 The Chinese Remainder Theorem
41(12)
3.1 Introduction
41(3)
3.2 The Chinese Remainder Theorem
44(3)
3.3 Worked-out examples
47(1)
3.4 Notes / Remarks
48(5)
Exercises
49(3)
References
52(1)
4 Mobius Inversion
53(16)
Introduction
53(1)
4.1 Abstract Mobius inversion
54(4)
4.2 Deduction: Mobius inversion of number theory
58(2)
4.3 The power set P(X) of a finite set X
60(2)
4.4 A worked-out example
62(1)
4.5 Notes / Remarks
63(6)
Exercises
64(4)
References
68(1)
5 Quadratic Residues (mod r) (r > 1)
69(14)
Introduction
69(1)
5.1 Preliminaries: Gauss' lemma
70(2)
5.2 Eisenstein lemma
72(3)
5.3 Quadratic reciprocity law
75(1)
5.4 First Supplement to quadratic reciprocity law
75(1)
5.5 Second supplement to quadratic reciprocity law
76(1)
5.6 The Jacobi symbol
76(1)
5.7 Worked-out examples
77(2)
5.8 Notes / Remarks
79(4)
Exercises
80(1)
References
81(2)
6 Decomposition of a Number as a Sum of Two or Four Squares
83(12)
Introduction
83(3)
6.1 Gaussian integers
86(1)
6.2 Integral quaternions
87(3)
6.3 Landau's Theorem
90(1)
6.4 Worked-out examples
90(2)
6.5 Notes / Remarks
92(3)
Exercises
93(1)
References
94(1)
7 Dirichlet Algebra of Arithmetical Functions
95(20)
Introduction
95(1)
7.1 Arithmetical convolutions
96(1)
7.2 Arithmetic functions
97(1)
7.3 Mobius inversion (another form)
98(2)
7.4 Unitary convolution
100(1)
7.5 UFD property of the ring of arithmetic functions
101(4)
7.6 Worked-out examples
105(3)
7.7 Notes / Remarks
108(7)
Exercises
110(2)
References
112(3)
8 Modular Arithmetical Functions
115(16)
Introduction
115(2)
8.1 Eckford Cohen's orthogonal property for Ramanujan sums
117(4)
8.2 Finite Fourier series representations of even functions (mod r)
121(3)
8.3 An application
124(1)
8.4 A worked-out example
125(1)
8.5 Notes / Remarks
126(5)
Exercises
128(1)
References
129(2)
9 A Generalization of Ramanujan Sums
131(14)
Introduction
131(1)
9.1 Jordan's totient Jk(r)
132(1)
9.2 Residue systems (mod k, r)
133(1)
9.3 A generalization of C(n, r)
134(3)
9.4 An application
137(1)
9.5 Worked-out examples
138(2)
9.6 Notes / Remarks
140(5)
Exercises
140(3)
References
143(2)
10 Ramanujan Expansions of Multiplicative Arithmetic Functions
145(16)
Introduction
145(3)
10.1 Averages of even functions (mod r)
148(3)
10.2 Series expansions
151(1)
10.3 Worked-out examples
152(2)
10.4 Notes / Remarks
154(7)
Exercises
156(2)
References
158(3)
Section II SELECTED TOPICS IN ALGEBRA
161(84)
11 On the Uniqueness of a Group of Order r(r > 1)
163(16)
Introduction
163(1)
11.1 On the nature of a group of order pq where p, q are primes (with p < q)
164(6)
11.2 Uniqueness of a group of order r
170(2)
11.3 A primality test
172(2)
11.4 A worked-out example
174(1)
11.5 A generalization
174(1)
11.6 Notes / Remarks
175(4)
Exercises
175(2)
References
177(2)
12 Quadratic Reciprocity in a Finite Group
179(16)
Introduction
179(1)
12.1 Preliminaries
179(3)
12.2 Group characters
182(4)
12.3 Quadratic reciprocity in respect of a finite group G
186(3)
12.4 A worked-out example
189(1)
12.5 Notes / Remarks
190(5)
Exercises
191(1)
References
192(3)
13 Commutative Rings with Unity
195(26)
Introduction
195(1)
13.1 Divisibility theory in integral domains
195(3)
13.2 Zorn's lemma
198(8)
13.3 Irreducibles and primes
206(2)
13.4 Euclidean domains
208(2)
13.5 Almost Euclidean domains
210(2)
13.6 Certain radicals of a ring / semisimplicity
212(2)
13.7 Worked-out examples
214(2)
13.8 Notes / Remarks
216(5)
Exercises
217(1)
References
218(3)
14 Noetherian and Artinian Rings
221(24)
Introduction
221(1)
14.1 Commutative rings with unity
222(1)
14.2 Properties of noetherian rings
223(6)
14.3 Lasker-Noether decomposition theorem
229(6)
14.4 Artinian rings
235(2)
14.5 Worked-out examples
237(3)
14.6 Notes / Remarks
240(5)
Exercises
241(2)
References
243(2)
Section III GLIMPSES OF THE THEORY OF ALGEBRAIC NUMBERS
245(54)
15 Dedekind Domains
247(26)
Introduction
247(1)
15.1 fl-modules
247(1)
15.2 Dedekind domains
248(11)
15.3 Elements integral over a ring R
259(3)
15.4 Integral domains having finite norm property
262(5)
15.5 Worked-out examples
267(1)
15.6 Notes / Remarks
268(5)
Exercises
269(3)
References
272(1)
16 Algebraic Number Fields
273(26)
Introduction
273(2)
16.1 Galois Theory for subfields of C
275(1)
16.2 The degree relation
276(2)
16.3 Algebraic numbers and algebraic number fields
278(2)
16.4 Algebraic integers
280(1)
16.5 The ideal class group
281(6)
16.6 The Diophantine equation x2 + 2y2 = n
287(1)
16.7 Finiteness of the class number
288(2)
16.8 Worked-out examples
290(3)
16.9 Notes / Remarks
293(6)
Exercises
294(2)
References
296(3)
Section IV SOME ADDITIONAL TOPICS
299(62)
17 Vaidyanathaswamy's Class-Division of Integers Modulo r
301(16)
Introduction
301(3)
17.1 An example [ 4] of class-division of integers (mod r)
304(1)
17.2 Evaluation of ykij
305(6)
17.3 An application
311(1)
17.4 A worked-out example
312(2)
17.5 Notes / Remarks
314(3)
Exercises
314(2)
References
316(1)
18 Burnside's Lemma and a Few of Its Applications
317(14)
Introduction
317(1)
18.1 Action of a group on a set
318(2)
18.2 Applications
320(5)
18.3 A worked-out example
325(2)
18.4 Notes / Remarks
327(4)
Exercises
327(2)
References
329(2)
19 On Cyclic Codes of Length n over Fq
331(20)
Introduction
331(2)
19.1 Mathematical formulation
333(1)
19.2 The binary symmetric channel
334(2)
19.3 Block codes
336(1)
19.4 Linear codes of length n over Fq
337(1)
19.5 Extension of Fields
338(2)
19.6 g-cyclotomic cosets mod n
340(1)
19.7 Cyclic codes of length n over Fq
341(1)
19.8 Factorization of xn - 1 (n ≤ 1)
342(2)
19.9 The generating polynomial of a cyclic code
344(2)
19.10 Worked-out examples
346(1)
19.11 Notes / Remarks
347(4)
Exercises
348(1)
References
349(2)
20 An Analogue of the Goldbach Problem
351(10)
Introduction
351(1)
20.1 The ring Mn(Z) of n × n matrices
352(3)
20.2 A matrix analogue of the Goldbach problem
355(2)
20.3 A worked-out example
357(1)
20.4 Notes / Remarks
357(4)
Exercises
358(1)
References
359(2)
Appendix A On the Partition Function p(r) (r ≥ 1)
361(2)
A.1 Definition and some properties
361(2)
References
362(1)
Appendix B Thumb-Nail Sketches of Biographies of Forty-One Prominent Mathematicians
363(30)
B.1 Euclid (circa 300 B.C)
363(1)
B.2 Eratosthenes (276-195/194 B.C)
364(1)
B.3 Diophantus (circa 250 A.D)
365(1)
B.4 Aryabhata (476-550 A.D)
366(1)
B.5 Brahmagupta (b. 598 A.D)
366(1)
B.6 Madhava(n) of Sangamagrama (circa 1100 A.D)
367(1)
B.7 Bhaskara II or Bhaskaracharya (Bhaskara, the learned) (1114--1185 A.D)
368(1)
B.8 Neelakanta Somayajin (1444--1544 A.D)
369(1)
B.9 Pierre de Fermat (1601--1665)
370(1)
B.10 Christian Goldbach (1690--1764)
370(1)
B.11 Leonhard Euler (1707--1783)
371(1)
B.12 Jean Le Rand d'Alembert (1717--1783)
371(1)
B.13 Joseph-Louis Lagrange (1736--1813)
372(1)
B.14 John Wilson (1741--1793)
373(1)
B.15 Adrien-Marie Legendre (1752--1833)
373(1)
B.16 Carl Friedrich Gauss (1777--1855)
374(1)
B.17 Niels Henrik Abel (1802--1829)
374(1)
B.18 Carl Gustav Jacob Jacobi (1804--1851)
374(1)
B.19 Johann Peter Gustav Lejeune Dirichlet (1805--1859)
375(1)
B.20 W. R. Hamilton (1805--1865)
376(1)
B.21 Eduard E. Kummer (1810--1893)
376(1)
B.22 Everiste Galois (1811--1832)
377(1)
B.23 Arthur Cayley (1821--1896)
377(1)
B.24 F. G. Max Eisenstein (1823--1852)
377(1)
B.25 Leopold Kronecker (1823--1891)
378(1)
B.26 Richard Dedekind (1831--1916)
378(2)
B.27 Peter Ludwig Mejdell Sylow (1832--1918)
380(1)
B.28 Edouard Lucas (1842--1891)
380(1)
B.29 Ferdinand Georg Frobenius (1849--1917)
381(1)
B.30 David Hilbert (1862--1943)
381(1)
B.31 Jacquess Hadamard (1865--1963)
382(1)
B.32 De la Vallee Poussin (1866--1962)
382(1)
B.33 Godfrey Herald Hardy (1877--1947)
383(1)
B.34 Emmy Noether (1882--1935)
384(1)
B.35 Srinivasa Ramanujan (1887--1920)
384(1)
B.36 R. Vaidyanathaswamy (1894--1960)
385(1)
B.37 Max Zorn (1906--1993)
386(1)
B.38 S. Minakshisundaram (1913--1968)
387(1)
B.39 Paul Erdos (1913--1996)
388(1)
B.40 C. S. Seshadri
388(1)
B.41 Herald Mead Stark
388(3)
References
389(2)
A Table Giving a Comparative Study of Number Theory and Algebra
391(2)
Appendix C Suggested for Further Study / Reading
393(6)
List of symbols
395(4)
Author Index 399(4)
Index of Mathematical Terms 403
Sivaramakrishnan Ramakrishna Ayyar has served a few educational institutions in Kerala as a math-teacher. In particular, he has served the university of Calicut, Kerala-673635 during the years 1977-1996.



A monograph entitled: Classical Theory of Arithmetic Functions (Marcel Dekker (1989)) was published during his visitorship (1987-88) at the University of Kansas, Lawrence, KS66045, U.S.A. He has also visited Mangalore University (1996-1997), Mangalagangotri, Mangalore-574199. He holds an Emeritus membership of American Mathematical Society, since 2008. His e-mail address is: rsjreeg@gmail.com.



The first edition of this book was published in the year 2006 (Chapman & Hall/CRC (Tayor & Francis Group)) when he could receive a Government of India Grant under Utilization of Services of Retired Scientists (USERS) scheme to work at the University of Calicut during the two-year period April, 2000March, 2002.
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