Introduction to Abstract Algebra, Solutions Manual

233 Pages·2012·12.4 MB·English

by  W. Keith Nicholson

Contents
Cover
Title Page
Copyright
Chapter 0: Preliminaries
0.1 Proofs
0.2 Sets
0.3 Mappings
0.4 Equivalences
Chapter 1: Integers and Permutations
1.1 Induction
1.2 Divisors and Prime Factorization
1.3 Integers Modulon
1.4 Permutations
Chapter 2: Groups
2.1 Binary Operations
2.2 Groups
2.3 Subgroups
2.4 Cyclic Groups and the Order of an Element
2.5 Homomorphisms and Isomorphisms
2.6 Cosets and Lagrange's Theorem
2.7 Groups of Motions and Symmetries
2.8 Normal Subgroups
2.9 Factor Groups
2
2.10 The Isomorphism Theorem
2.11 An Application to Binary Linear Codes
Chapter 3: Rings
3.1 Examples and Basic Properties
3.2 Integral Domains and Fields
3.3 Ideals and Factor Rings
3.4 Homomorphisms
3.5 Ordered Integral Domains
Chapter 4: Polynomials
4.1 Polynomials
4.2 Factorization of Polynomials over a Field
4.3 Factor Rings of Polynomials over a Field
4.4 Partial Fractions
4.5 Symmetric Polynomials
Chapter 5: Factorization in Integral Domains
5.1 Irreducibles and Unique Factorization
5.2 Principal Ideal Domains
Chapter 6: Fields
6.1 Vector Spaces
6.2 Algebraic Extensions
6.3 Splitting Fields
6.4 Finite Fields
6.5 Geometric Constructions
6.7 An Application to Cyclic and BCH Codes
3
Chapter 7: Modules over Principal Ideal Domains
7.1 Modules
7.2 Modules over a Principal Ideal Domain
Chapter 8:p-Groups and the Sylow Theorems
8.1 Products and Factors
8.2 Cauchy's Theorem
8.3 Group Actions
8.4 The Sylow Theorems
8.5 Semidirect Products
8.6 An Application to Combinatorics
Chapter 9: Series of Subgroups
9.1 The Jordan-Hölder Theorem
9.2 Solvable Groups
9.3 Nilpotent Groups
Chapter 10: Galois Theory
10.1 Galois Groups and Separability
10.2 The Main Theorem of Galois Theory
10.3 Insolvability of Polynomials
10.4 Cyclotomic Polynomials and Wedderburn's Theorem
Chapter 11: Finiteness Conditions for Rings and Modules
11.1 Wedderburn's Theorem
11.2 The Wedderburn-Artin Theorem
Appendices
Appendix A: Complex Numbers
4
Appendix B: Matrix Arithmetic
Appendix C: Zorn's Lemma
5
6
Copyright 2012 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
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7
Library of Congress Cataloging-in-Publication Data:
Nicholson, W. Keith.
Introduction to abstract algebra / W. Keith Nicholson. – 4th ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-118-28815-3 (cloth)
1. Algebra, Abstract. I. Title.
QA162.N53 2012
512'.02–dc23 2011031416
8
Chapter 0
Preliminaries
0.1 Proofs
1.
a.
2 2 2
1.Ifn=2k,kaninteger,thenn =(2k) =4k isamultiple
of 4.
2
2.Theconverseistrue:Ifn isamultipleof4thennmust
2
be even becausen is odd whennis odd (Example 1).
c.
3 2 3 2
1.Verify:2 −6·2 +11·2−6=0and3 −6·3 +11·
3 − 6 = 0.
2. The converse is false: x = 1 is a counterexample.
because
2.
a. Either n = 2k or n = 2k + 1, for some integer k. In the first
2 2 2 2
casen = 4k ; in the secondn = 4(k +k) + 1.
3 3
c.Ifn= 3k, thenn −n= 3(9k −k); ifn= 3k+ 1, then
3 3 2
ifn= 3k+ 2, thenn −n= 3(9k + 18k + 11k+ 2).
3.
a.
1. If n is not odd, then n = 2k, k an integer, k ≥ 1, so n is
not a prime.
2. The converse is false: n = 9 is a counterexample; it is
odd but is not a prime.
c.
1. If then , that is a > b,
contrary to the assumption.
9
2. The converse is true: If then
, that isa≤b.
4.
a. If x > 0 and y > 0 assume .
Squaring gives , whence
.Thismeansxy=0sox=0ory=0,contradicting
our assumption.
c. Assume all have birthdays in different months. Then there
can be at most 12 people, one for each month, contrary to
hypothesis.
5.
2
a. n = 11 is a counterexample because then n + n + 11 = 11 ·
2
13isnotprime.Notethatn +n+11isprimeif1≤n≤9asis
2
readily verified, but n = 10 is also a counterexample as 10 +
2
10 + 11 = 11 .
c.n=6isacounterexample becausetherearethen31regions.
Note that the result holds if 2 ≤n≤ 5.
0.2 Sets
1.
a.A= {x x= 5k, ,k≥ 1}
2.
a.{1, 3, 5, 7, . . . }
c.{ − 1, 1, 3}
e.{ } = ∅ is the empty set by Example 3.
3.
a.Not equal: −1 Abut −1 ∉B.
c.Equal to {a,l,o,y}.
e.Not equal: 0 Abut 0 ∉B.
g.Equal to { − 1, 0, 1}.
4.
a.∅, {2}
10