
- •List of Symbols
- •Classical Algebra
- •Modern Algebra
- •Binary Operations
- •Algebraic Structures
- •Extending Number Systems
- •Algebra of Sets
- •Number of Elements in a Set
- •Boolean Algebras
- •Propositional Logic
- •Switching Circuits
- •Divisors
- •Posets and Lattices
- •Normal Forms and Simplification of Circuits
- •Transistor Gates
- •Representation Theorem
- •Exercises
- •Groups and Symmetries
- •Subgroups
- •Cyclic Groups and Dihedral Groups
- •Morphisms
- •Permutation Groups
- •Even and Odd Permutations
- •Equivalence Relations
- •Normal Subgroups and Quotient Groups
- •Morphism Theorem
- •Direct Products
- •Groups of Low Order
- •Action of a Group on a Set
- •Exercises
- •Translations and the Euclidean Group
- •Matrix Groups
- •Finite Groups in Two Dimensions
- •Proper Rotations of Regular Solids
- •Finite Rotation Groups in Three Dimensions
- •Necklace Problems
- •Coloring Polyhedra
- •Counting Switching Circuits
- •Exercises
- •Monoids and Semigroups
- •Finite-State Machines
- •Quotient Monoids and the Monoid of a Machine
- •Exercises
- •Rings
- •Integral Domains and Fields
- •Subrings and Morphisms of Rings
- •New Rings From Old
- •Field of Fractions
- •Convolution Fractions
- •Exercises
- •Euclidean Rings
- •Euclidean Algorithm
- •Unique Factorization
- •Factoring Real and Complex Polynomials
- •Factoring Rational and Integral Polynomials
- •Factoring Polynomials over Finite Fields
- •Linear Congruences and the Chinese Remainder Theorem
- •Exercises
- •Ideals and Quotient Rings
- •Computations in Quotient Rings
- •Morphism Theorem
- •Quotient Polynomial Rings that are Fields
- •Exercises
- •Field Extensions
- •Algebraic Numbers
- •Galois Fields
- •Primitive Elements
- •Exercises
- •Latin Squares
- •Orthogonal Latin Squares
- •Finite Geometries
- •Magic Squares
- •Exercises
- •Constructible Numbers
- •Duplicating a Cube
- •Trisecting an Angle
- •Squaring the Circle
- •Constructing Regular Polygons
- •Nonconstructible Number of Degree 4
- •Exercises
- •The Coding Problem
- •Simple Codes
- •Polynomial Representation
- •Matrix Representation
- •Error Correcting and Decoding
- •BCH Codes
- •Exercises
- •Induction
- •Divisors
- •Prime Factorization
- •Proofs in Mathematics
- •Modern Algebra in General
- •History of Modern Algebra
- •Connections to Computer Science and Combinatorics
- •Groups and Symmetry
- •Rings and Fields
- •Convolution Fractions
- •Latin Squares
- •Geometrical Constructions
- •Coding Theory
- •Chapter 2
- •Chapter 3
- •Chapter 4
- •Chapter 5
- •Chapter 6
- •Chapter 7
- •Chapter 8
- •Chapter 9
- •Chapter 10
- •Chapter 11
- •Chapter 12
- •Chapter 13
- •Chapter 14
- •Index

BIBLIOGRAPHY AND
REFERENCES
Proofs in Mathematics
1.Bloch, Ethan D., Proofs and Fundamentals: A First Course in Abstract Mathematics. Boston: Birkhauser, 2000.
2.Schumacher, Carol, Chapter Zero: Fundamental Notions of Abstract Mathematics, 2nd ed. Reading, Mass.: Addison-Wesley, 2000.
3.Solow, Daniel, How to Read and Do Proofs: An Introduction to Mathematical Thought Processes, 3rd ed. New York: Wiley, 2002.
Modern Algebra in General
4.Artin, Michael, Algebra. Upper Saddle River, N.J.: Prentice Hall, 1991.
5.Birkhoff, Garrett, and Thomas C. Bartee, Modern Applied Algebra. New York: McGraw-Hill, 1970.
6.Birkhoff, Garrett, and Saunders Maclane, A Survey of Modern Algebra, 4th ed. New York: Macmillan, 1977.
7.Durbin, John R., Modern Algebra: An Introduction, 4th ed. New York: Wiley, 2000.
8.Gallian, Joseph A., Contemporary Abstract Algebra, 5th ed. Boston: Houghton Mifflin, 2002.
9.Herstein, I. N., Topics in Algebra, 2nd ed. New York: Wiley, 1973.
10.Lidl, Rudolf, and Gunter Pilz, Applied Abstract Algebra, 2nd ed. New York: Springer-Verlag, 1997.
11.Nicholson, W. Keith, Introduction to Abstract Algebra, 2nd ed. New York: Wiley, 1999.
12.Weiss, Edwin, First Course in Algebra and Number Theory. San Diego, Calif.: Academic Press, 1971.
History of Modern Algebra
13.Kline, Morris, Mathematical Thought from Ancient to Modern Times, Vol. 3. New York: Oxford University Press, 1990 (Chap. 49).
14.Stillwell, John, Mathematics and Its History, 2nd ed. New York: Springer-Verlag, 2002.
Modern Algebra with Applications, Second Edition, by William J. Gilbert and W. Keith Nicholson ISBN 0-471-41451-4 Copyright 2004 John Wiley & Sons, Inc.
BIBLIOGRAPHY AND REFERENCES |
307 |
Connections to Computer Science and Combinatorics
15.Biggs, Norman L., Discrete Mathematics, 2nd ed. Oxford: Oxford University Press, 2003.
16.Davey, B. A., and H. A. Priestley, Introduction to Lattices and Order, 2nd ed. Cambridge: Cambridge University Press, 2002.
17.Gathen, Joachim von zur, and Jurgen¨ Gerhard, Modern Computer Algebra, 2nd ed. Cambridge: Cambridge University Press, 2003.
18.Hopcroft, John E., Rajeev Motwani, and Jeffrey D. Ullman, Introduction to Automata Theory, Languages, and Computation, 2nd ed. Reading, Mass.: Addison-Wesley, 2000.
19.Knuth, Donald E., The Art of Computer Programming, Vol. 2, Seminumerical Algorithms, 3rd ed. Reading, Mass.: Addison-Wesley, 1998.
20.Kolman, Bernard, Robert C. Busby, and Sharon Cutler Ross, Discrete Mathematical Structures, 4th ed. Upper Saddle River, N.J.: Prentice Hall, 1999.
21.Mendelson, Elliott, Schaum’s Outline of Theory and Problems of Boolean Algebra and Switching Circuits. New York: McGraw-Hill, 1970.
22.Stone, Harold S., Discrete Mathematical Structures and Their Applications. Chicago: Science Research Associates, 1973.
23.Whitesitt, J. Eldon, Boolean Algebra and Its Applications. New York: Dover, 1995.
Groups and Symmetry
24.Armstrong, Mark Anthony, Groups and Symmetry. New York: Springer-Verlag, 1988.
25.Baumslag, Benjamin, and Bruce Chandler, Schaum’s Outline of Group Theory. New York: McGraw-Hill, 1968.
26.Budden, F. J., The Fascination of Groups. Cambridge: Cambridge University Press, 1972.
27.Coxeter, H. S. M., Introduction to Geometry, 2nd ed. New York: Wiley, 1989.
28.Cundy, H. Martyn, and A. P. Rollett, Mathematical Models, 3rd ed. Stradbroke, Norfolk, England: Tarquin, 1981.
29.Field, Michael, and Martin Golubitsky, Symmetry in Chaos: A Search for Pattern in Mathematics, Art and Nature. Oxford: Oxford University Press, 1992.
30.Hall, Marshall, Jr., The Theory of Groups. New York: Macmillan, 1959 (reprinted by the American Mathematical Society, 1999).
31.Lomont, John S., Applications of Finite Groups. New York: Dover, 1993.
32.Shapiro, Louis W., Finite groups acting on sets with applications. Mathematics Magazine, 46 (1973), 136–147.
Rings and Fields
33.Cohn, P. M., Introduction to Ring Theory. New York: Springer-Verlag, 2000.
34.Lidl, Rudolf, and Harald Niederreiter, Introduction to Finite Fields and Their Applications, rev. ed. Cambridge: Cambridge University Press, 1994.
35.Stewart, Ian, Galois Theory, 3rd ed. Boca Raton, Fla.: CRC Press, 2003.
Convolution Fractions
36.Erdelyi, Arthur, Operational Calculus and Generalized Functions. New York: Holt, Rinehart and Winston, 1962.
37.Marchand, Jean Paul, Distributions: An Outline. Amsterdam: North-Holland, 1962.
Latin Squares
38.Ball, W. W. Rouse, and H. S. M. Coxeter, Mathematical Recreations and Essays. New York: Dover, 1987.
308 |
BIBLIOGRAPHY AND REFERENCES |
39.Lam, C. W. H., The search for a finite projective plane of order 10. American Mathematical Monthly, 98(1991), 305–318.
40.Laywine, Charles F., and Gary L. Mullen, Discrete Mathematics Using Latin Squares. New York: Wiley, 1998.
Geometrical Constructions
41.Courant, Richard, Herbert Robbins, and Ian Stewart, What Is Mathematics? New York: Oxford University Press, 1996.
42.Kalmanson, Kenneth, A familiar constructibility criterion. American Mathematical Monthly, 79(1972), 277–278.
43.Kazarinoff, Nicholas D., Ruler and the Round. New York: Dover, 2003.
44.Klein, Felix, Famous Problems of Elementary Geometry. New York: Dover, 1956.
Coding Theory
45.Kirtland, Joseph, Identification Numbers and Check Digit Schemes. Washington, D.C.: Mathematical Association of America, 2001.
46.Roman, Steven, Introduction to Coding and Information Theory. New York: Springer-Verlag, 1997.

ANSWERS TO THE
ODD-NUMBERED
EXERCISES
CHAPTER 2 |
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2.1. Always true. |
2.3. When A ∩ (B C) = . |
2.5. When A ∩ (B C) = . |
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2.13.|A B C D| = |A| + |B| + |C| + |D| − |A ∩ B| −|A ∩ C| − |A ∩ D| − |B ∩ C| − |B ∩ D| − |C ∩ D|
+|A ∩ B ∩ C| + |A ∩ B ∩ D| + |A ∩ C ∩ D| + |B ∩ C ∩ D|
−|A ∩ B ∩ C ∩ D|. |
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2.17. Yes; P( ). |
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2.15. 4. |
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2.25. |
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(a) and (b) are equivalent and (c) and (d) are equivalent.
2.27.(a) is a contradiction and (b), (c) and (d) are tautologies.
2.29.
B A C′
D
B′ C′
B′
2.31.B C .
2.33. A (B A ); (A B) (A B ) (A B); A B.
Modern Algebra with Applications, Second Edition, by William J. Gilbert and W. Keith Nicholson ISBN 0-471-41451-4 Copyright 2004 John Wiley & Sons, Inc.