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    Nova Science Publishers, Inc. Huntington, NY

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    Library of Congress Cataloging-in-Publication Data

    Badawi, Ayman.

    Abstract algebra manual: problems and solutions / Ayman Badawi

    p. cm.

    Includes bibliographical references and index.

    1. Algebra, Abstract. I. Title.

    QA162 .B33 2001



    Copyright 2001 by Nova Science Publishers, Inc.

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    with or arising out of information contained in this book.

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    A Abelian, 5

    automorphisms, 2

    B Binomial Expansion Theorem, 68

    C cancelation, left-, 32

    cancelation, right-, 32

    centralizer, 1

    closed, 3

    closure, 11

    common divisor, greatest, 1

    common divisor, least, 1

    common entries, 4

    complex numbers, all nonzero, 29

    contradiction, 11

    cosets, 1

    cosets, left, 21

    cycle, even, 19

    cycles, disjoint, 4, 18

    cyclic group, infinite, 35

    cyclic, 3

    D distinct powers, 3

    distinct prime numbers, 3

    divisor, 3

    divisor, positive, 4

    domain, 62

    domain, integral, 65

    domain, principal ideal, 69

    E element, 2

    element, nilpotent, 62

    element, non-identity, 8, 49

    elements, distinct, 3, 13

    elements, nilpotent, 55

    F field, 56

    field, extension, 58, 93

    field, finite extension, 58

    field, finite, 59

    field, infinite, 85

    field, some extension, 96

    field, splitting, 92

    finite group, 4

  • A. Badawi 108

    finite index, 29

    finite set, 2

    finite, order, 3

    G generators, 16

    group homomorphism, 5, 34

    group homomorphism, nontrivial, 37

    group isomorphism, 36

    group, Abelian, 13

    group, cyclic, 3

    group, finite Abelian, 36

    group, infinite cyclic, 16

    group, infinite, 17

    group, non-Abelian, 24, 47

    group, non-cyclic Abelian, 17

    group, non-cyclic, 25

    group, unit, 2

    group-axioms, 11

    H homomorphism, ring, 34

    I ideal , maximal, 67

    ideal domain, principle, 56

    ideal, nonzero prime, 88

    ideal, prime, 65

    ideal, principal, 67

    ideal, proper, 66

    ideals, all prime (maximal), 88

    ideals, prime (maximal), 81

    idempotent, 62

    identity matrix, 63

    infinite order, 3, 15, 85

    integer, even, 70

    integer, odd, 70

    integer, positive, 4

    integers, 27

    integral domain, 55, 73

    integral domain, finite 73

    integer, positive, 2, 31

    integers, 2

    isomorphic, 2

    isomorphisms, 2

    L least common multiple, 4

    linear factors, 90

    M matrix, square, 2

    matrix-multiplication, 2

    maximal ideal, 56, 57

    maximal order, 53

    module, multiplication, 2

    module, intergers, 2

    multiplicative identity, 77

    multiplicity, 86

    N nilpotent, 63

    non-Albelian, 45

    nonzero-ring homomorphism, 83

    normalizer, 1

    number, prime, 2, 23

    numbers, all rational, 55

    numbers, all real, 55

    numbers, complex, 2

    numbers, rational, 2

    O ONTO, 80,

    P permutation of a finite set, 4

    permutations, 2

    polynomial, 55

    polynomial, irreducible, 59, 90

  • Abstract Algebra Manual: Problems and Solutions 109

    polynomial, nonzero, 56

    polynomials, monic irreducible, 57, 102

    prime number, 5, 8

    R rational numbers, all nonzero, 28

    rational numbers, all positive, 28

    real number, positive, 44

    real numbers, all nonzero negative, 29

    real numbers, all nonzero positive, 29

    relatively prime, 3

    ring homomorphism, 56, 80

    ring isomorphism, 77

    ring, commutative, 55, 65

    ring, finite commutative, 73

    ring, finite, 74

    ring, non-commutative, 61

    root, common, 99

    root, multiple, 99

    S subgroup of finite index, proper, 28

    subgroup, 1

    subgroup, Abelian normal, 33

    subgroup, cyclic, 2

    subgroup, finite, 17

    subgroup, normal cyclic, 30

    subgroup, normal, 1, 5, 26

    subgroup, proper, 22

    subgroup, smallest, 15

    subgroup, Sylow p-, 5

    subgroup, unique, 4, 18

    subgroups, proper, 17

    subring isomorphic, 79

    subring, 68

    Sylow p-subgroups, 52

    Sylow q-subgroups, 48

    U unit, 74

    Z zero (root), 56

    zero divisor, 74

    zero divisors, 55

    ABSTRACT ALGEBRA MANUAL: PROBLEMS AND SOLUTIONSABSTRACT ALGEBRA MANUAL: PROBLEMS AND SOLUTIONSLibrary of Congress Cataloging-in-Publication DataContents IntroductionChapter 1: Tools and Major Results of Groups1.1 Notations1.2 Results

    Chapter 2: Problems in Group Theory2.1 Elementary Properties of Groups 2.2 Subgroups2.3 Cyclic Groups2.4 Permutation Groups2.5 Cosets and Lagrange's Theorem2.6 Normal Subgroups and Factor Groups2.7 Group Homomorphisms and Direct Product 2.8 Sylow Theorems

    Chapter 3: Tools and Major Results of Ring Theory3.1 Notations3.2 Major Results of Ring Theory

    Chapter 4: Problems in Ring Theory4.1 Basic Properties of Rings 4.2 Ideals, Subrings, and Factor Rings4.3 Integral Domains, and Zero Divisors 4.4 Ring Homomorphisms and Ideals4.5 Polynomial Rings4.6 Factorization in Polynomial Rings4.7 Extension Fields, and Algebraic Fields 4.8 Finite Fields