It is common knowledge among mathematicians that much of modern algebra has its roots in the issue of solvability of equations by radicals, and this book succeeds at providing an introduction to modern algebra while keeping this relationship in view at all times.  Most modern algebra books employ an axiomatic strategy that begins with abstract groups and ends with fields, ignoring the issue of solvability of equations by radicals.  By contract, this book follows the paper trail from the Renaissance solution of the cubic equation to Galois's description of his ideas.  In the process, all the important concepts are encountered, each in a well-motivated manner.  This Second Edition features a new chapter introducing rings and fields as well as many new sections on topics such as group homomorphisms, the RSA algorithm, complex conugation, the factorization of real polynomials, and the fundamental theorem of algebra.  New appendices are included on topics such as logic and proof, sets, functions, and equivalence relations, a ring without prime factorizations, and Sylow theory and classification of groups.  The author provides readers with a unique opportunity to view the evolution of modern algebra as a consistent movement from concrete problems to abstract principles. By including several pertinent excerpts from the writings of mathematicians whose works kept the movement going, he helps readers experience the drama of discovery behind the formulation of pivotal ideas. Readers also develop a more immediate and well-grounded understanding of how equations lead to permutation groups and what those groups can inform us about multivariate functions and the 15-puzzle. Chapter coverage includes: The Early History; Complex Numbers; Solutions of Equations; Modular Arithmetic; The Binomial Theorem and Modular Powers; Introduction to Rings and Fields; Polynomials Over a Field; Galois Fields; Permutations; Groups; Quotient Groups and Their Uses; and Topics in Elementary Group Theory.