This text introduces students to the algebraic concepts of group and rings, providing a comprehensive discussion of theory as well as a significant number of applications for each.
• Comprehensive coverage of abstract algebra – Includes discussions of the fundamental theorem of Galois theory; Jordan-Holder theorem; unitriangular groups; solvable groups; construction of free groups; von Dyck's theorem, and presentations of groups by generators and relations.
• Significant applications for both group and commutative ring theories, especially with Gr ö bner bases – Helps students see the immediate value of abstract algebra.
• Flexible presentation – May be used to present both ring and group theory in one semester, or for two-semester course in abstract algebra.
• Number theory – Presents concepts such as induction, factorization into primes, binomial coefficients and DeMoivre's Theorem, so students can learn to write proofs in a familiar context.
• Section on Euclidean rings – Demonstrates that the quotient and remainder from the division algorithm in the Gaussian integers may not be unique. Also, Fermat's Two-Squares theorem is proved.
• Sylow theorems – Discusses the existence of Sylow subgroups as well as conjugacy and the congruence condition on their number.
• Fundamental theorem of finite abelian groups – Covers the basis theorem as well as the uniqueness to isomorphism
• Extensive references and consistent numbering system for lemmas, theorems, propositions, corollaries, and examples – Clearly organized notations, hints, and appendices simplify student reference.
