Algebra: Groups, Rings, and Fields

Algebra is a subject we have become acquainted with during most of our mathematical education, often in connection with the solution of equations. Algebra: Groups, Rings, and Fields, Second Edition deals with developments related to their solutions. The principle at the heart of abstract algebra, a subject that enables one to deduce sweeping conclusions from elementary premises, is that the process of abstraction enables us to solve a variety of such problems with economy of effort. This leads to the glorious world of mathematical discovery. This second edition follows the original three-pronged approach: the theory of finite groups, number theory, and Galois’ amazing theory of field extensions tying solvability of equations to group theory. As algebra has branched out in many directions, the authors strive to keep the text manageable while at the same time introducing the student to exciting new paths. In order to support this approach, the authors broadened the first edition, giving monoids a greater role, and relying more on matrices. Hundreds of new exercises were added. A course in abstract algebra, properly presented, could treat mathematics as an art as well as a science. In this exposition, we try to present underlying ideas, as well as the results they yield.

Table of Contents:
1 Monoids and Groups 1.1 Examples of Groups and MonoidsWhen Is a Monoid a Group? 1.2 Exercises 2 Lagrange’s Theorem, Cosets, and an Application to Number Theory 2.1 Cosets 2.2 Fermat’s Little Theorem 2.3 Exercises 3 Cauchy’s Theorem: Showing that a Number Is Greater Than 1 3.1 The Exponent 3.2 The symmetric group Sn: Our Main Example 3.3 The Product of Two Subgroups 3.4 Exercises 4 Structure of Groups: Homomorphisms, Isomorphisms, and Invariants 4.1 Homomorphic Images 4.2 Exercises 5 Normal Subgroups: The Building Blocks of the Structure Theory 5.1 The Residue Group 5.2 Noether’s Isomorphism Theorems 5.3 Conjugates in Sn 5.4 The Alternating Group 5.5 Exercises 6 Classifying Groups: Cyclic Groups and Direct Products 6.1 Cyclic Groups 6.2 Generators of a Group 6.3 Direct Products 6.4 Application: Some Algebraic Cryptosystems 6.5 Exercises 7 Finite Abelian Groups 7.1 Abelian p-Groups 7.2 Proof of the Fundamental Theorem for Finite abelian Groups 7.3 The Classification of Finite abelian Groups 7.4 Exercises 8 Generators and Relations 8.1 Description of Groups of Low Order 8.3 Exercises 9 When Is a Group a Group? (Cayley’s Theorem) 9.1 The Generalized Cayley Theorem 9.2 Introduction to Group Representations 9.3 Exercises 10 Conjugacy Classes and the Class Equation 10.1 The Center of a Group 10.2 Exercises 11 Sylow Subgroups 11.1 Groups of Order Less Than 60 11.2 Finite Simple Groups 11.3 Exercises 12 Solvable Groups: What Could Be Simpler? 12.1 Commutators 12.2 Solvable Groups 12.3 Automorphisms of Groups 12.4 Exercises 13 Groups of Matrices 13.1 Exercises 14 An Introduction to Rings 14.1 Domains and Skew Fields 14.2 Left Ideals 14.3 Exercises 15 The Structure Theory of Rings 15.1 Ideals 15.2 Noether’s Isomorphism Theorems for Rings 15.3 Exercises 16 The Field of Fractions: A Study in Generalization 16.1 Intermediate Rings 16.2 Exercises 17 Polynomials and Euclidean Domains 17.1 The Ring of Polynomials 17.2 Euclidean Domains 17.3 Unique Factorization 17.4 Exercises 18 Principal Ideal Domains: Induction without Numbers 18.1 Prime Ideals 18.2 Noetherian RingsExercises 19 Roots of Polynomials 19.1 Finite Subgroups of Fields 19.2 Primitive Roots of 1 19.3 Exercises 20 Applications: Famous Results from Number Theory 20.1 A Theorem of Fermat 20.2 Addendum: “Fermat’s Last Theorem” 20.3 Exercises 21 Irreducible Polynomials 21.1 Polynomials over UFDs 21.2 Eisenstein’s Criterion 21.3 Exercises 22 Field Extensions: Creating Roots of Polynomials 22.1 Algebraic Elements 22.2 Finite Field Extensions 22.3 Exercises 23 The Geometric Problems of Antiquity 23.1 Construction by Straight Edge and Compass 23.2 Algebraic Description of Constructibility 23.3 Solution of the Geometric Problems of Antiquity 23.4 Exercises 24 Adjoining Roots to Polynomials: Splitting Fields 24.1 Splitting Fields 24.2 Separable Polynomials and Separable Extensions 24.3 Exercises 25 Finite Fields 25.1 Uniqueness 25.2 Existence 25.3 Exercises 26 The Galois Correspondence 26.1 The Galois Group of a Field Extension 26.2 The Galois Group and Intermediate Fields 26.3 Exercises 27 Applications of the Galois Correspondence 27.1 Finite Separable Field Extensions and the Normal Closure 27.2 The Galois Group of a Polynomial 27.3 Constructible n-gons 27.4 Finite Fields 27.5 The Fundamental Theorem of Algebra 27.6 Exercises 28 Solving Equations by Radicals 28.1 Radical Extensions 28.2 Solvable Galois Groups 28.3 Computing the Galois Group 28.4 Exercises 29 Integral Extensions 29.1 Exercises 30 Group Representations and their Characters 30.1 Exercises 31 Transcendental Numbers: e and π 31.1 Transcendence of e 31.2 Transcendence of π 32 Skew Field Theory 32.1 The Quaternion Algebra 32.2 Polynomials over Skew Fields 32.3 Structure Theorems for Skew Fields 32.4 Exercises 33 Where Do We Go From Here? 33.1 Modules 33.2 Matrix Algebras and their Substructures 33.3 Nonassociative Rings and Algebras 33.4 Hyperfields 33.5 Exercises