book

Abstract Algebra, 2nd Edition

Name: Abstract Algebra, 2nd Edition
ISBN: 9781003814184

by Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom

December 2023

Intermediate to advanced

547 pages

15h 6m

English

Chapman and Hall/CRC

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Cover Page
Half-Title Page
Series Page
Title Page
Copyright Page
Contents
Note to Students
Preface
I Number Systems
1 The Integers
IntroductionArithmetic and Ordering AxiomsDivisibility in ZCongruenceFactoring, Prime Numbers, and Greatest Common DivisorsLinear CombinationsProofs of the Division Algorithm and the Fundamental Theorem of ArithmeticConcluding ActivitiesExercises

2 Equivalence Relations and Zn
Congruence ClassesEquivalence RelationsEquivalence ClassesThe Number System ZnBinary OperationsZero Divisors and Units in ZnConcluding ActivitiesExercises
3 Algebra in Other Number Systems
IntroductionSubsets of the Real NumbersThe Complex NumbersMatricesCollections of SetsPutting It All TogetherConcluding ActivitiesExercises
II Rings
4 Introduction to Rings
IntroductionBasic Properties of RingsCommutative Rings and Rings with IdentityUniqueness of Identities and InversesZero Divisors and Multiplicative CancellationFields and Integral DomainsConcluding ActivitiesExercisesConnections
5 Integer Multiples and Exponents
IntroductionInteger Multiplication and ExponentiationNonpositive Multiples and ExponentsProperties of Integer Multiplication and ExponentiationThe Characteristic of a RingConcluding ActivitiesExercisesConnections
6 Subrings, Extensions, and Direct Sums
IntroductionThe Subring TestSubfields and Field ExtensionsDirect SumsConcluding ActivitiesExercisesConnections
7 Isomorphism and Invariants
IntroductionIsomorphisms of RingsRenaming ElementsPreserving OperationsProving IsomorphismWell-Defined FunctionsDisproving IsomorphismInvariantsConcluding ActivitiesExercisesConnections
III Polynomial Rings
8 Polynomial Rings
Polynomials over Commutative RingsPolynomials over an Integral DomainPolynomial FunctionsConcluding ActivitiesExercisesConnections
9 Divisibility in Polynomial Rings
IntroductionThe Division Algorithm in F[x]Greatest Common Divisors of PolynomialsRelatively Prime PolynomialsThe Euclidean Algorithm for PolynomialsConcluding ActivitiesExercisesConnections
10 Roots, Factors, and Irreducible Polynomials
Polynomial Functions and RemaindersRoots of Polynomials and the Factor TheoremIrreducible PolynomialsUnique Factorization in F[x]Concluding ActivitiesExercisesConnections
11 Irreducible Polynomials
IntroductionFactorization in C[x]Factorization in R[x]Factorization in Q[x]Polynomials with No Linear Factors in Q[x]Reducing Polynomials in Z[x] Modulo PrimesEisenstein's CriterionFactorization in F[x] for Other Fields FSummaryConcluding ActivitiesExercisesConnections
12 Quotients of Polynomial Rings
IntroductionCongruence Modulo a PolynomialCongruence Classes of PolynomialsThe Set F[x]/fxSpecial Quotients of Polynomial RingsAlgebraic NumbersConcluding ActivitiesExercisesConnections
IV More Ring Theory
13 Ideals and Homomorphisms
IntroductionIdealsCongruence Modulo an IdealMaximal and Prime IdealsHomomorphismsThe Kernel and Image of a HomomorphismThe First Isomorphism Theorem for RingsConcluding ActivitiesExercisesConnections
14 Divisibility and Factorization in Integral Domains
IntroductionDivisibility and Euclidean DomainsPrimes and IrreduciblesUnique Factorization DomainsProof 1: Generalizing Greatest Common DivisorsProof 2: Principal Ideal DomainsConcluding ActivitiesExercisesConnections
15 From Z to C
IntroductionFrom W to ZOrdered RingsFrom Z to QOrdering on QFrom Q to RFrom R to CA Characterization of the IntegersConcluding ActivitiesExercisesConnections
V Groups
16 Symmetry
IntroductionSymmetriesSymmetries of Regular PolygonsConcluding ActivitiesExercisesConnections
17 An Introduction to Groups
GroupsExamples of GroupsBasic Properties of GroupsIdentities and Inverses in a GroupThe Order of a GroupGroups of UnitsConcluding ActivitiesExercisesConnections
18 Integer Powers of Elements in a Group
IntroductionPowers of Elements in a GroupConcluding ActivitiesExercisesConnections
19 Subgroups
IntroductionThe Subgroup TestThe Center of a GroupThe Subgroup Generated by an ElementConcluding ActivitiesExercisesConnections
20 Subgroups of Cyclic Groups
IntroductionSubgroups of Cyclic GroupsProperties of the Order of an ElementFinite Cyclic GroupsInfinite Cyclic GroupsConcluding ActivitiesExercisesConnections
21 The Dihedral Groups
IntroductionRelationships between Elements in DnGenerators and Group PresentationsConcluding ActivitiesExercisesConnections
22 The Symmetric Groups
IntroductionThe Symmetric Group of a SetPermutation Notation and CyclesThe Cycle Decomposition of a PermutationTranspositionsEven and Odd Permutations and the Alternating GroupConcluding ActivitiesExercisesConnections
23 Cosets and Lagrange's Theorem
IntroductionA Relation in GroupsCosetsLagrange's TheoremConcluding ActivitiesExercisesConnections
24 Normal Subgroups and Quotient Groups
IntroductionAn Operation on CosetsNormal SubgroupsQuotient GroupsCauchy's Theorem for Finite Abelian GroupsSimple Groups and the Simplicity of AnConcluding ActivitiesExercisesConnections
25 Products of Groups
External Direct Products of GroupsOrders of Elements in Direct ProductsInternal Direct Products in GroupsConcluding ActivitiesExercisesConnections
26 Group Isomorphisms and Invariants
IntroductionIsomorphisms of GroupsRenaming ElementsPreserving OperationsProving IsomorphismSome Basic Properties of IsomorphismsWell-Defined FunctionsDisproving IsomorphismInvariantsIsomorphism ClassesIsomorphisms and Cyclic GroupsCayley's TheoremConcluding ActivitiesExercisesConnections
27 Homomorphisms and Isomorphism Theorems
HomomorphismsThe Kernel of a HomomorphismThe Image of a HomomorphismThe Isomorphism Theorems for GroupsThe First Isomorphism Theorem for GroupsThe Second Isomorphism Theorem for GroupsThe Third Isomorphism Theorem for GroupsThe Fourth Isomorphism Theorem for GroupsConcluding ActivitiesExercisesConnections
28 The Fundamental Theorem of Finite Abelian Groups
IntroductionThe Components: p-GroupsThe Fundamental TheoremConcluding ActivitiesExercisesConnections
29 The First Sylow Theorem
IntroductionConjugacy and the Class EquationThe Class EquationCauchy's TheoremThe First Sylow TheoremThe Second and Third Sylow TheoremsConcluding ActivitiesExercisesConnections
30 The First Sylow Theorem
IntroductionConjugate Subgroups and NormalizersThe Second Sylow TheoremThe Third Sylow TheoremConcluding ActivitiesExercisesConnections
VI Fields and Galois Theory
31 Finite Fields, the Group of Units in Zn, and Splitting Fields
IntroductionFinite FieldsThe Group of Units of a Finite FieldThe Group of Units of ZnSplitting FieldsConcluding ActivitiesExercisesConnections
32 Extensions of Fields
IntroductionA Quick Review of Linear AlgebraExtension Fields and the Degree of an ExtensionField AutomorphismsConcluding ActivitiesExercisesConnections
33 Galois Theory
IntroductionThe Galois GroupThe Fundamental Theorem of Galois TheorySolvability by RadicalsSolvable GroupsPolynomials Not Solvable By RadicalsConcluding ActivitiesExercisesConnections
Index

Content preview from Abstract Algebra, 2nd Edition

Investigation 18 Integer Powers of Elements in a Group

Our study of groups began in Investigation 16, where we learned about the set of symmetries of an object. We then saw how many familiar sets, like $Z$ , $Z_{n}$ , and sets of invertible square matrices, all had a structure that was similar to that of a set of symmetries—namely, the structure of a group. One of the defining axioms ...

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Abstract Algebra, 2nd Edition

by Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom

Investigation 18 Integer Powers of Elements in a Group

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