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Abstract Algebra

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Table of Contents

The Integers The Integers: An Introduction Introduction Integer Arithmetic Ordering Axioms What's Next Concluding Activities Exercises Divisibility of Integers Introduction Quotients and Remainders TheWell-Ordering Principle Proving the Division Algorithm Putting It All Together Congruence Concluding Activities Exercises Greatest Common Divisors Introduction Calculating Greatest Common Divisors The Euclidean Algorithm GCDs and Linear Combinations Well-Ordering, GCDs, and Linear Combinations Concluding Activities Exercises Prime Factorization Introduction Defining Prime The Fundamental Theorem of Arithmetic Proving Existence Proving Uniqueness Putting It All Together Primes and Irreducibles in Other Number Systems Concluding Activities Exercises Other Number Systems Equivalence Relations and Zn Congruence Classes Equivalence Relations Equivalence Classes The Number System Zn Binary Operations Zero Divisors and Units in Zn Concluding Activities Exercises Algebra Introduction Subsets of the Real Numbers The Complex Numbers Matrices Collections of Sets Putting It All Together Concluding Activities Exercises Rings An Introduction to Rings Introduction Basic Properties of Rings Commutative Rings and Rings with Identity Uniqueness of Identities and Inverses Zero Divisors and Multiplicative Cancellation Fields and Integral Domains Concluding Activities Exercises Connections Integer Multiples and Exponents Introduction Integer Multiplication and Exponentiation Nonpositive Multiples and Exponents Properties of Integer Multiplication and Exponentiation The Characteristic of a Ring Concluding Activities Exercises Connections Subrings, Extensions, and Direct Sums Introduction The Subring Test Subfields and Field Extensions Direct Sums Concluding Activities Exercises Connections Isomorphism and Invariants Introduction Isomorphisms of Rings Proving Isomorphism Disproving Isomorphism Invariants Concluding Activities Exercises Connections Polynomial Rings Polynomial Rings Polynomial Rings Polynomials over an Integral Domain Polynomial Functions Concluding Activities Exercises Connections Appendix - Proof that R[x] Is a Commutative Ring Divisibility in Polynomial Rings Introduction The Division Algorithm in F[x] Greatest Common Divisors of Polynomials Relatively Prime Polynomials The Euclidean Algorithm for Polynomials Concluding Activities Exercises Connections Roots, Factors, and Irreducible Polynomials Polynomial Functions and Remainders Roots of Polynomials and the Factor Theorem Irreducible Polynomials Unique Factorization in F[x] Concluding Activities Exercises Connections Irreducible Polynomials Introduction Factorization 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 Primes Eisenstein's Criterion Factorization in F[x] for Other Fields F Summary The Cubic Formula Concluding Activities Exercises Appendix - Proof of the Fundamental Theorem of Algebra Quotients of Polynomial Rings Introduction CongruenceModulo a Polynomial Congruence Classes of Polynomials The Set F[x]/hf(x)i Special Quotients of Polynomial Rings Algebraic Numbers Concluding Activities Exercises Connections More Ring Theory Ideals and Homomorphisms Introduction Ideals CongruenceModulo an Ideal Maximal and Prime Ideals Homomorphisms The Kernel and Image of a Homomorphism The First Isomorphism Theorem for Rings Concluding Activities Exercises Connections Divisibility and Factorization in Integral Domains Introduction Divisibility and Euclidean Domains Primes and Irreducibles Unique Factorization Domains Proof 1: Generalizing Greatest Common Divisors Proof 2: Principal Ideal Domains Concluding Activities Exercises Connections From Z to C Introduction FromW to Z Ordered Rings From Z to Q Ordering on Q From Q to R From R to C A Characterization of the Integers Concluding Activities Exercises Connections VI Groups 269 Symmetry Introduction Symmetries Symmetries of Regular Polygons Concluding Activities Exercises An Introduction to Groups Groups Examples of Groups Basic Properties of Groups Identities and Inverses in a Group The Order of a Group Groups of Units Concluding Activities Exercises Connections Integer Powers of Elements in a Group Introduction Powers of Elements in a Group Concluding Activities Exercises Connections Subgroups Introduction The Subgroup Test The Center of a Group The Subgroup Generated by an Element Concluding Activities Exercises Connections Subgroups of Cyclic Groups Introduction Subgroups of Cyclic Groups Properties of the Order of an Element Finite Cyclic Groups Infinite Cyclic Groups Concluding Activities Exercises The Dihedral Groups Introduction Relationships between Elements in Dn Generators and Group Presentations Concluding Activities Exercises Connections The Symmetric Groups Introduction The Symmetric Group of a Set Permutation Notation and Cycles The Cycle Decomposition of a Permutation Transpositions Even and Odd Permutations and the Alternating Group Concluding Activities Exercises Connections Cosets and Lagrange's Theorem Introduction A Relation in Groups Cosets Lagrange's Theorem Concluding Activities Exercises Connections Normal Subgroups and Quotient Groups Introduction An Operation on Cosets Normal Subgroups Quotient Groups Cauchy's Theorem for Finite Abelian Groups Simple Groups and the Simplicity of An Concluding Activities Exercises Connections Products of Groups External Direct Products of Groups Orders of Elements in Direct Products Internal Direct Products in Groups Concluding Activities Exercises Connections Group Isomorphisms and Invariants Introduction Isomorphisms of Groups Proving Isomorphism Some Basic Properties of Isomorphisms Well-Defined Functions Disproving Isomorphism Invariants Isomorphism Classes Isomorphisms and Cyclic Groups Cayley's Theorem Concluding Activities Exercises Connections Homomorphisms and Isomorphism Theorems Homomorphisms The Kernel of a Homomorphism The Image of a Homomorphism The Isomorphism Theorems for Groups Concluding Activities Exercises Connections The Fundamental Theorem of Finite Abelian Groups Introduction The Components: p-Groups The Fundamental Theorem Concluding Activities Exercises The First Sylow Theorem Introduction Conjugacy and the Class Equation Cauchy's Theorem The First Sylow Theorem The Second and Third Sylow Theorems Concluding Activities Exercises Connections The Second and Third Sylow Theorems Introduction Conjugate Subgroups and Normalizers The Second Sylow Theorem The Third Sylow Theorem Concluding Activities Exercises Special Topics RSA Encryption Introduction Congruence and Modular Arithmetic The Basics of RSA Encryption An Example Why RSA Works Concluding Thoughts and Notes Exercises Check Digits Introduction Check Digits Credit Card Check Digits ISBN Check Digits Verhoeff's Dihedral Group D5 Check Concluding Activities Exercises Connections Games: NIM and the 15 Puzzle The Game of NIM The 15 Puzzle Concluding Activities Exercises Connections Finite Fields, the Group of Units in Zn, and Splitting Fields Introduction Finite Fields The Group of Units of a Finite Field The Group of Units of Zn Splitting Fields Concluding Activities Exercises Connections Groups of Order 8 and 12: Semidirect Products of Groups Introduction Groups of Order 8 Semi-direct Products of Groups Groups of Order 12 and p3 Concluding Activities Exercises Connections Appendices Functions Special Types of Functions: Injections and Surjections Composition of Functions Inverse Functions Theorems about Inverse Functions Concluding Activities Exercises Mathematical Induction and the Well-Ordering Principle Introduction The Principle of Mathematical Induction The Extended Principle of Mathematical Induction The Strong Form of Mathematical Induction TheWell-Ordering Principle The Equivalence of the Well-Ordering Principle and the Principles of Mathematical Induction. Concluding Activities Exercises


"This book arose from the authors' approach to teaching abstract algebra. They place an emphasis on active learning and on developing students' intuition through their investigation of examples. ... The text is organized in such a way that it is possible to begin with either rings or groups." -Florentina Chirtes, Zentralblatt MATH 1295

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