{"product_id":"introduction-to-abstract-algebra-9781421411767","title":"Introduction to Abstract Algebra","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003ePresents an approach to teach one of math's most intimidating concepts. This book allows beginner-level students to follow the progression from familiar topics such as rings, numbers, and groups to more difficult concepts.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003eA friendly introduction to the subject-no sentence is terse, and in fact useful additional statements are sprinkled throughout an argument... Suitably chosen examples are given throughout the text to illustrate the definitions, proofs and arguments, and there are also plenty of exercises at the end of each chapter for the reinforcement of understanding. It is an excellent text for a university course. -- Peter Shiu Mathematical Gazette The utmost detailed presentation of the core material, the wealth of illustrating examples, and the many outlooks for further study make this excellent algebra primer a highly welcome, useful and valuable addition to the abundant textbook literature in the field. Zentralblatt Math\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003cp\u003ePreface\u003cbr\u003e1. Abstract Algebra and Algebraic Reasoning\u003cbr\u003e1.1. Abstract Algebra\u003cbr\u003e1.2. Algebraic Structures\u003cbr\u003e1.3. The Algebraic Method\u003cbr\u003e1.4. The Standard Number Systems\u003cbr\u003e1.5. The Integers and Induction\u003cbr\u003e1.6. Exercises\u003cbr\u003e2. Algebraic Preliminaries\u003cbr\u003e2.1. Sets and Set Theory\u003cbr\u003e2.1.1. Set Operations\u003cbr\u003e2.2. Functions\u003cbr\u003e2.3. Equivalence Relations and Factor Sets\u003cbr\u003e2.4. Sizes of Sets\u003cbr\u003e2.5. Binary Operations\u003cbr\u003e2.5.1. The Algebra of Sets\u003cbr\u003e2.6. Algebraic Structures and Isomorphisms\u003cbr\u003e2.7. Groups\u003cbr\u003e2.8. Exercises\u003cbr\u003e3. Rings and the Integers\u003cbr\u003e3.1. Rings and the Ring of Integers\u003cbr\u003e3.2. Some Basic Properties of Rings and Subrings\u003cbr\u003e3.3. Examples of Rings\u003cbr\u003e3.3.1. The Modular Rings: The Integers Modulo n\u003cbr\u003e3.3.2. Noncommutative Rings\u003cbr\u003e3.3.3. Rings Without Identities\u003cbr\u003e3.3.4. Rings of Subsets: Boolean Rings\u003cbr\u003e3.3.5. Direct Sums of Rings\u003cbr\u003e3.3.6. Summary of Examples\u003cbr\u003e3.4. Ring Homomorphisms and Isomorphisms\u003cbr\u003e3.5. Integral Domains and Ordering\u003cbr\u003e3.6. Mathematical Induction and the Uniqueness of Z\u003cbr\u003e3.7. Exercises\u003cbr\u003e4. Number Theory and Unique Factorization\u003cbr\u003e4.1. Elementary Number Theory\u003cbr\u003e4.2. Divisibility and Primes\u003cbr\u003e4.3. Greatest Common Divisors\u003cbr\u003e4.4. The Fundamental Theorem of Arithmetic\u003cbr\u003e4.5. Congruences and Modular Arithmetic\u003cbr\u003e4.6. Unique Factorization Domains\u003cbr\u003e4.7. Exercises\u003cbr\u003e5. Fields: The Rationals, Reals and Complexes\u003cbr\u003e5.1. Fields and Division Rings\u003cbr\u003e5.2. Construction and Uniqueness of the Rationals\u003cbr\u003e5.2.1. Fields of Fractions\u003cbr\u003e5.3. The Real Number System\u003cbr\u003e5.3.1. The Completeness of R (Optional)\u003cbr\u003e5.3.2. Characterization of R (Optional)\u003cbr\u003e5.3.3. The Construction of R (Optional)\u003cbr\u003e5.3.4. The p-adic Numbers (Optional)\u003cbr\u003e5.4. The Field of Complex Numbers\u003cbr\u003e5.4.1. Geometric Interpretation\u003cbr\u003e5.4.2. Polar Form and Euler's Identity\u003cbr\u003e5.4.3. DeMoivre's Theorem for Powers and Roots\u003cbr\u003e5.5. Exercises\u003cbr\u003e6. Basic Group Theory\u003cbr\u003e6.1. Groups, Subgroups and Isomorphisms\u003cbr\u003e6.2. Examples of Groups\u003cbr\u003e6.2.1. Permutations and the Symmetric Group\u003cbr\u003e6.2.2. Examples of Groups: Geometric Transformation Groups\u003cbr\u003e6.3. Subgroups and Lagrange's Theorem\u003cbr\u003e6.4. Generators and Cyclic Groups\u003cbr\u003e6.5. Exercises\u003cbr\u003e7. Factor Groups and the Group Isomorphism Theorems\u003cbr\u003e7.1. Normal Subgroups\u003cbr\u003e7.2. Factor Groups\u003cbr\u003e7.2.1. Examples of Factor Groups\u003cbr\u003e7.3. The Group Isomorphism Theorems\u003cbr\u003e7.4. Exercises\u003cbr\u003e8. Direct Products and Abelian Groups\u003cbr\u003e8.1. Direct Products of Groups\u003cbr\u003e8.1.1. Direct Products of Two Groups\u003cbr\u003e8.1.2. Direct Products of Any Finite Number of Groups\u003cbr\u003e8.2. Abelian Groups\u003cbr\u003e8.2.1. Finite Abelian Groups\u003cbr\u003e8.2.2. Free Abelian Groups\u003cbr\u003e8.2.3. The Basis Theorem for Finitely Generated Abelian Groups\u003cbr\u003e8.3. Exercises\u003cbr\u003e9. Symmetric and Alternating Groups\u003cbr\u003e9.1. Symmetric Groups and Cycle Structure\u003cbr\u003e9.1.1. The Alternating Groups\u003cbr\u003e9.1.2. Conjugation in Sn\u003cbr\u003e9.2. The Simplicity of An\u003cbr\u003e9.3. Exercises\u003cbr\u003e10. Group Actions and Topics in Group Theory\u003cbr\u003e10.1. Group Actions\u003cbr\u003e10.2. Conjugacy Classes and the Class Equation\u003cbr\u003e10.3. The Sylow Theorems\u003cbr\u003e10.3.1. Some Applications of the Sylow Theorems\u003cbr\u003e10.4. Groups of Small Order\u003cbr\u003e10.5. Solvability and Solvable Groups\u003cbr\u003e10.5.1. Solvable Groups\u003cbr\u003e10.5.2. The Derived Series\u003cbr\u003e10.6. Composition Series and the Jordan-Holder Theorem\u003cbr\u003e10.7. Exercises\u003cbr\u003e11. Topics in Ring Theory\u003cbr\u003e11.1. Ideals in Rings\u003cbr\u003e11.2. Factor Rings and the Ring Isomorphism Theorem\u003cbr\u003e11.3. Prime and Maximal Ideals\u003cbr\u003e11.3.1. Prime Ideals and Integral Domains\u003cbr\u003e11.3.2. Maximal Ideals and Fields\u003cbr\u003e11.4. Principal Ideal Domains and Unique Factorization\u003cbr\u003e11.5. Exercises\u003cbr\u003e12. Polynomials and Polynomial Rings\u003cbr\u003e12.1. Polynomials and Polynomial Rings\u003cbr\u003e12.2. Polynomial Rings over a Field\u003cbr\u003e12.2.1. Unique Factorization of Polynomials\u003cbr\u003e12.2.2. Euclidean Domains\u003cbr\u003e12.2.3. F[x] as a Principal Ideal Domain\u003cbr\u003e12.2.4. Polynomial Rings over Integral Domains\u003cbr\u003e12.3. Zeros of Polynomials\u003cbr\u003e12.3.1. Real and Complex Polynomials\u003cbr\u003e12.3.2. The Fundamental Theorem of Algebra\u003cbr\u003e12.3.3. The Rational Roots Theorem\u003cbr\u003e12.3.4. Solvability by Radicals\u003cbr\u003e12.3.5. Algebraic and Transcendental Numbers\u003cbr\u003e12.4. Unique Factorization in Z[x]\u003cbr\u003e12.5. Exercises\u003cbr\u003e13. Algebraic Linear Algebra\u003cbr\u003e13.1. Linear Algebra\u003cbr\u003e13.1.1. Vector Analysis in R3\u003cbr\u003e13.1.2. Matrices and Matrix Algebra\u003cbr\u003e13.1.3. Systems of Linear Equations\u003cbr\u003e13.1.4. Determinants\u003cbr\u003e13.2. Vector Spaces over a Field\u003cbr\u003e13.2.1. Euclidean n-Space\u003cbr\u003e13.2.2. Vector Spaces\u003cbr\u003e13.2.3. Subspaces\u003cbr\u003e13.2.4. Bases and Dimension\u003cbr\u003e13.2.5. Testing for Bases in Fn\u003cbr\u003e13.3. Dimension and Subspaces\u003cbr\u003e13.4. Algebras\u003cbr\u003e13.5. Inner Product Spaces\u003cbr\u003e13.5.1. Banach and Hilbert Spaces\u003cbr\u003e13.5.2. The Gram-Schmidt Process and Orthonormal Bases\u003cbr\u003e13.5.3. The Closest Vector Theorem\u003cbr\u003e13.5.4. Least-Squares Approximation\u003cbr\u003e13.6. Linear Transformations and Matrices\u003cbr\u003e13.6.1. Matrix of a Linear Transformation\u003cbr\u003e13.6.2. Linear Operators and Linear Functionals\u003cbr\u003e13.7. Exercises\u003cbr\u003e14. Fields and Field Extensions\u003cbr\u003e14.1. Abstract Algebra and Galois Theory\u003cbr\u003e14.2. Field Extensions\u003cbr\u003e14.3. Algebraic Field Extensions\u003cbr\u003e14.4. F-automorphisms, Conjugates and Algebraic Closures\u003cbr\u003e14.5. Adjoining Roots to Fields\u003cbr\u003e14.6. Splitting Fields and Algebraic Closures\u003cbr\u003e14.7. Automorphisms and Fixed Fields\u003cbr\u003e14.8. Finite Fields\u003cbr\u003e14.9. Transcendental Extensions\u003cbr\u003e14.10. Exercises\u003cbr\u003e15. A Survey of Galois Theory\u003cbr\u003e15.1. An Overview of Galois Theory\u003cbr\u003e15.2. Galois Extensions\u003cbr\u003e15.3. Automorphisms and the Galois Group\u003cbr\u003e15.4. The Fundamental Theorem of Galois Theory\u003cbr\u003e15.5. A Proof of the Fundamental Theorem of Algebra\u003cbr\u003e15.6. Some Applications of Galois Theory\u003cbr\u003e15.6.1. The Insolvability of the Quintic\u003cbr\u003e15.6.2. Some Ruler and Compass Constructions\u003cbr\u003e15.6.3. Algebraic Extensions of R\u003cbr\u003e15.7. Exercises\u003cbr\u003eBibliography\u003cbr\u003eIndex\u003c\/p\u003e","brand":"Johns Hopkins University Press","offers":[{"title":"Default Title","offer_id":49529528615255,"sku":"9781421411767","price":71.82,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781421411767.jpg?v=1731875970","url":"https:\/\/bookcurl.com\/products\/introduction-to-abstract-algebra-9781421411767","provider":"Book Curl","version":"1.0","type":"link"}