Abstract algebra : an inquiry based approach /: an inquiry based approach. (2013)

• Record Type: Book
• Further Information: Note: By Jonathan K. Hodge, Steven Schlicker, Theodore A. Sundstrom.
• Authors: Hodge, Jonathan K, 1980-; Schlicker, Steven, 1958-; Sundstrom, Theodore A
• 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&lt
• Publisher Details: Place of publication not identified : Chapman and Hall/CRC
• Publication Date: 2013
• Extent: 1 online resource (595 pages), (52 illustrations)
• Subjects: 512.02; Algebra, Abstract
• Languages: English
• ISBNs: 9781466567085; 1466567082
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.