Introduction to Abstract Algebra, Second Edition presents abstract
algebra as the main tool underlying discrete mathematics and the
digital world. It avoids the usual groups first/rings first dilemma
by introducing semigroups and monoids, the multiplicative
structures of rings, along with groups. This new edition of a
widely adopted textbook covers applications from biology, science,
and engineering. It offers numerous updates based on feedback from
first edition adopters, as well as improved and simplified proofs
of a number of important theorems. Many new exercises have been
added, while new study projects examine skewfields, quaternions,
and octonions. The first three chapters of the book show how
functional composition, cycle notation for permutations, and matrix
notation for linear functions provide techniques for practical
computation. These three chapters provide a quick introduction to
algebra, sufficient to exhibit irrational numbers or to gain a
taste of cryptography. Chapters four through seven cover abstract
groups and monoids, orthogonal groups, stochastic matrices,
Lagrange's theorem, groups of units of monoids, homomorphisms,
rings, and integral domains. The first seven chapters provide basic
coverage of abstract algebra, suitable for a one-semester or
two-quarter course. Each chapter includes exercises of varying
levels of difficulty, chapter notes that point out variations in
notation and approach, and study projects that cover an array of
applications and developments of the theory. The final chapters
deal with slightly more advanced topics, suitable for a
second-semester or third-quarter course. These chapters delve
deeper into the theory of rings, fields, and groups. They discuss
modules, including vector spaces and abelian groups, group theory,
and quasigroups. This textbook is suitable for use in an
undergraduate course on abstract algebra for mathematics, computer
science, and education majors, along with students from other STEM
fields.
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