Description
Book SynopsisProvides a readable, student-friendly, and somewhat sophisticated introduction to abstract algebra. It is aimed at sophomore or junior undergraduates who are seeing the material for the first time. In addition to the usual definitions and theorems, there is ample discussion to help students build intuition and learn how to think about the abstract concepts.
Trade Review“Written with great care and clarity, Shahriari's
Algebra in Action provides an excellent introduction to abstract algebra. I have used the book twice to teach abstract algebra class at Reed College, and it's a perfect fit. The book is sophisticated yet readable, and packed with examples and exercises. Group actions appear early on, serving to motivate and unify many of the important concepts in group theory. The book also includes plenty of material on rings and fields, including the basics of Galois theory.” — Jamie Pommersheim, Reed College
“The structure of the text
Algebra in Action lets students see what groups really do right from the very beginning. In the very first chapter, the author introduces a rich selection of examples, the dihedral groups, the symmetric group, the integers modulo n, and matrix groups, that students can see 'in action' before the presentation of the formal definitions of groups and group actions in chapter 2 where the theoretical foundations are introduced. Students return to these examples again and again as the formal theory unfolds, seeing how the theory lets them study all groups at once...It is one of the few texts at the undergraduate level that supports the incorporation of group actions at an early stage in the course.” — Jessica Sidman, Mount Holyoke College
“It is rigorous, well-written, ample in terms of problems and solutions provided, and sufficiently advanced for its target audience.” — Jason M. Graham,
MAA ReviewsTable of Contents
- (Mostly finite) group theory: Four basic examples
- Groups: The basics
- The alternating groups
- Group actions
- A subgroup acts on the group: Cosets and Lagrange's theorem
- A group acts on itself: Counting and the conjugation of action
- Acting on subsets, cosets, and subgroups: The Sylow theorems
- Counting the number of orbits
- The lattice of subgroups
- Acting on its subgroups: Normal subgroups and quotient groups
- Group homomorphisms
- Using Sylow theorems to analyze finite groups
- Direct and semidirect products
- Solvable and nilpotent groups
- (Mostly commutative) ring theory: Rings
- Homomorphisms, ideals, and quotient rings
- Field of fractions and localization
- Factorization, EDs, PIDs, and UFDs
- Polynomial rings
- Gaussian integers and (a little) number theory
- Field and Galois theory: Introducing field theory and Galois theory
- Field extensions
- Straightedge and compass constructions
- Splitting fields and Galois groups
- Galois, normal, and separable extensions
- Fundamental theorem of Galois theory
- Finite fields and cyclotomic extensions
- Radical extensions, solvable groups, and the quintic
- Hints for selected problems
- Short answers for selected problems
- Complete solutions for selected (odd-numbered) problems
- Bibliography
- Index
