Certain Number-Theoretic Episodes In Algebra, Second Edition (Chapman & Hall/CRC Pure and Applied Mathematics)

Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
Acknowledgment
About the Author
Chapter-Wise Description of the Contents
Section I: - ELEMENTS OF THE THEORY OF NUMBERS
	1: From Euclid to Lucas: Elementary Theorems Revisited
		Introduction
		1.1. The quotient ring Z/rZ (r > 1)
		1.2. Congruences modulo a prime
		1.3. Fermat’s two-squares theorem
		1.4. Lagrange’s four-squares theorem
		1.5. Worked-out examples
		1.6. Notes / Remarks
		Exercises
		References
	2: Solutions of Congruences, Primitive Roots
		Introduction
		2.1. Theorems on congruences
		2.2. Worked-out examples
		2.3. Notes / Remarks
		Exercises
		References
	3: The Chinese Remainder Theorem
		3.1. Introduction
		3.2. The Chinese Remainder Theorem
		3.3. Worked-out examples
		3.4. Notes / Remarks
		Exercises
		References
	4: Möbius Inversion
		Introduction
		4.1. Abstract Möbius inversion
		4.2. Deduction: Möbius inversion of number theory
		4.3. The power set P(X) of a finite set X
		4.4. A worked-out example
		4.5. Notes / Remarks
		Exercises
		References
	5: Quadratic Residues (mod r) (r > 1)
		Introduction
		5.1. Preliminaries: Gauss’ lemma
		5.2. Eisenstein lemma
		5.3. Quadratic reciprocity law
		5.4. First Supplement to quadratic reciprocity law
		5.5. Second supplement to quadratic reciprocity law
		5.6. The Jacobi symbol
		5.7. Worked-out examples
		5.8. Notes / Remarks
		Exercises
		References
	6: Decomposition of a Number as a Sum of Two or Four Squares
		Introduction
		6.1. Gaussian integers
		6.2. Integral quaternions
		6.3. Landau’s Theorem
		6.4. Worked-out examples
		6.5. Notes / Remarks
		Exercises
		References
	7: Dirichlet Algebra of Arithmetical Functions
		Introduction
		7.1. Arithmetical convolutions
		7.2. Arithmetic functions
		7.3. Möbius inversion (another form)
		7.4. Unitary convolution
		7.5. UFD property of the ring of arithmetic functions
		7.6. Worked-out examples
		7.7. Notes / Remarks
		Exercises
		References
	8: Modular Arithmetical Functions
		Introduction
		8.1. Eckford Cohen’s orthogonal property for Ramanujan sums
		8.2. Finite Fourier series representations of even functions (mod r)
		8.3. An application
		8.4. A worked-out example
		8.5. Notes / Remarks
		Exercises
		References
	9: A Generalization of Ramanujan Sums
		Introduction
		9.1. Jordan’s totient Jk(r)
		9.2. Residue systems (mod k, r)
		9.3. A generalization of C(n, r)
		9.4. An application
		9.5. Worked-out examples
		9.6. Notes / Remarks
		Exercises
		References
	10: Ramanujan Expansions of Multiplicative Arithmetic Functions
		Introduction
		10.1. Averages of even functions (mod r)
		10.2. Series expansions
		10.3. Worked-out examples
		10.4. Notes / Remarks
		Exercises
		References
Section II: - SELECTED TOPICS IN ALGEBRA
	11: On the Uniqueness of a Group of Order r (r > 1)
		Introduction
		11.1. On the nature of a group of order pq where p, q are primes (with p < q)
		11.2. Uniqueness of a group of order r
		11.3. A primality test
		11.4. A worked-out example
		11.5. A generalization
		11.6. Notes / Remarks
		Exercises
		References
	12: Quadratic Reciprocity in a Finite Group
		Introduction
		12.1. Preliminaries
		12.2. Group characters
		12.3. Quadratic reciprocity in respect of a finite group G
		12.4. A worked-out example
		12.5. Notes / Remarks
		Exercises
		References
	13: Commutative Rings with Unity
		Introduction
		13.1. Divisibility theory in integral domains
		13.2. Zorn’s lemma
		13.3. Irreducibles and primes
		13.4. Euclidean domains
		13.5. Almost Euclidean domains
		13.6. Certain radicals of a ring / semisimplicity
		13.7. Worked-out examples
		13.8. Notes / Remarks
		Exercises
		References
	14: Noetherian and Artinian Rings
		Introduction
		14.1. Commutative rings with unity
		14.2. Properties of noetherian rings
		14.3. Lasker-Noether decomposition theorem
		14.4. Artinian rings
		14.5. Worked-out examples
		14.6. Notes / Remarks
		Exercises
		References
Section III: - GLIMPSES OF THE THEORY OF ALGEBRAIC NUMBERS
	15: Dedekind Domains
		Introduction
		15.1. R-modules
		15.2. Dedekind domains
		15.3. Elements integral over a ring R
		15.4. Integral domains having finite norm property
		15.5. Worked-out examples
		15.6. Notes / Remarks
		Exercises
		References
	16: Algebraic Number Fields
		Introduction
		16.1. Galois Theory for subfields of C
		16.2. The degree relation
		16.3. Algebraic numbers and algebraic number fields
		16.4. Algebraic integers
		16.5. The ideal class group
		16.6. The Diophantine equation x2 + 2y2 = n
		16.7. Finiteness of the class number
		16.8. Worked-out examples
		16.9. Notes / Remarks
		Exercises
		References
Section IV: - SOME ADDITIONAL TOPICS
	17: Vaidyanathaswamy’s Class-Division of Integers Modulo r
		Introduction
		17.1. An example [4] of class-division of integers (mod r)
		17.2. Evaluation of γkij
		17.3. An application
		17.4. A worked-out example
		17.5. Notes / Remarks
		Exercises
		References
	18: Burnside’s Lemma and a Few of Its Applications
		Introduction
		18.1. Action of a group on a set
		18.2. Applications
		18.3. A worked-out example
		18.4. Notes / Remarks
		Exercises
		References
	19: On Cyclic Codes of Length n over Fq
		Introduction
		19.1. Mathematical formulation
		19.2. The binary symmetric channel
		19.3. Block codes
		19.4. Linear codes of length n over Fq
		19.5. Extension of Fields
		19.6. q-cyclotomic cosets mod n
		19.7. Cyclic codes of length n over Fq
		19.8. Factorization of xn - 1 (n ≥ 1)
		19.9. The generating polynomial of a cyclic code
		19.10. Worked-out examples
		19.11. Notes / Remarks
		Exercises
		References
	20: An Analogue of the Goldbach Problem
		Introduction
		20.1. The ring Mn(Z) of n × n matrices
		20.2. A matrix analogue of the Goldbach problem
		20.3. A worked-out example
		20.4. Notes / Remarks
		Exercises
		References
	Appendix A: On the Partition Function p(r) (r ≥ 1)
		A.1. Definition and some properties
		References
	Appendix B: Thumb-Nail Sketches of Biographies of Forty-One Prominent Mathematicians
		B.1. Euclid (circa 300 B.C)
		B.2. Eratosthenes (276–195/194 B.C)
		B.3. Diophantus (circa 250 A.D)
		B.4. Aryabhata (476–550 A.D)
		B.5. Brahmagupta (b. 598 A.D)
		B.6. Madhava(n) of Sangamagrāma (circa 1100 A.D)
		B.7. Bhaskara II or Bhaskaracharya (Bhaskara, the learned) (1114–1185 A.D)
		B.8. Neelakanta Somayajin (1444–1544 A.D)
		B.9. Pierre de Fermat (1601–1665)
		B.10. Christian Goldbach (1690–1764)
		B.11. Leonhard Euler (1707–1783)
		B.12. Jean Le Rand d’Alembert (1717–1783)
		B.13. Joseph-Louis Lagrange (1736–1813)
		B.14. John Wilson (1741–1793)
		B.15. Adrien-Marie Legendre (1752–1833)
		B.16. Carl Friedrich Gauss (1777–1855)
		B.17. Niels Henrik Abel (1802–1829)
		B.18. Carl Gustav Jacob Jacobi (1804–1851)
		B.19. Johann Peter Gustav Lejeune Dirichlet (1805–1859)
		B.20. W. R. Hamilton (1805–1865)
		B.21. Eduard E. Kummer (1810–1893)
		B.22. Everiste Galois (1811–1832)
		B.23. Arthur Cayley (1821–1896)
		B.24. F. G. Max Eisenstein (1823–1852)
		B.25. Leopold Kronecker (1823–1891)
		B.26. Richard Dedekind (1831–1916)
		B.27. Peter Ludwig Mejdell Sylow (1832–1918)
		B.28. Edouard Lucas (1842–1891)
		B.29. Ferdinand Georg Fröbenius (1849–1917)
		B.30. David Hilbert (1862–1943)
		B.31. Jacquess Hadamard (1865–1963)
		B.32. De la Vallee Poussin (1866–1962)
		B.33. Godfrey Herald Hardy (1877–1947)
		B.34. Emmy Noether (1882–1935)
		B.35. Srinivasa Ramanujan (1887–1920)
		B.36. R. Vaidyanathaswamy (1894–1960)
		B.37. Max Zorn (1906–1993)
		B.38. S. Minakshisundaram (1913–1968)
		B.39. Paul Erdos (1913–1996)
		B.40. C. S. Seshadri
		B.41. Herald Mead Stark
		References
	A Table Giving a Comparative Study of Number Theory and Algebra
	Appendix C: Suggested for Further Study / Reading
List of symbols
Author Index
Index of Mathematical Terms