ALGEBRA IN ACTION: A COURSE IN GROUPS, RINGS, AND FIELDS

ALGEBRA IN ACTION: A COURSE IN GROUPS, RINGS, AND FIELDS

Editorial:
AMS (AMERICAN MATHEMATICAL SOCIETY)
Año de edición:
Materia
Matematicas
ISBN:
978-1-4704-2849-5
Páginas:
655
N. de edición:
1
Idioma:
Inglés
Disponibilidad:
Disponible en 2-3 semanas

Descuento:

-5%

Antes:

136,00 €

Despues:

129,20 €

• (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, & 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

This text—based on the author's popular courses at Pomona College—provides 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. The book has over 1300 exercises and mini-projects of varying degrees of difficulty, and, to facilitate active learning and self-study, hints and short answers for many of the problems are provided. There are full solutions to over 100 problems in order to augment the text and to model the writing of solutions. Lattice diagrams are used throughout to visually demonstrate results and proof techniques. The book covers groups, rings, and fields. In group theory, group actions are the unifying theme and are introduced early. Ring theory is motivated by what is needed for solving Diophantine equations, and, in field theory, Galois theory and the solvability of polynomials take center stage. In each area, the text goes deep enough to demonstrate the power of abstract thinking and to convince the reader that the subject is full of unexpected results.

Author
Shahriar Shahriari: Pomona College, Claremont, CA