Algebra in Action

Shahriar Shahriari, Pomona College, Claremont, CA.

• (Mostly finite) group theory: Four basic examplesGroups: The basicsThe alternating groupsGroup actionsA subgroup acts on the group: Cosets and Lagrange's theoremA group acts on itself: Counting and the conjugation of actionActing on subsets, cosets, and subgroups: The Sylow theoremsCounting the number of orbitsThe lattice of subgroupsActing on its subgroups: Normal subgroups and quotient groupsGroup homomorphismsUsing Sylow theorems to analyze finite groupsDirect and semidirect productsSolvable and nilpotent groups(Mostly commutative) ring theory: RingsHomomorphisms, ideals, and quotient ringsField of fractions and localizationFactorization, EDs, PIDs, and UFDsPolynomial ringsGaussian integers and (a little) number theoryField and Galois theory: Introducing field theory and Galois theoryField extensionsStraightedge and compass constructionsSplitting fields and Galois groupsGalois, normal, and separable extensionsFundamental theorem of Galois theoryFinite fields and cyclotomic extensionsRadical extensions, solvable groups, and the quinticHints for selected problemsShort answers for selected problemsComplete solutions for selected (odd-numbered) problemsBibliographyIndex

“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 Reviews