Fields and Galois Theory

This book provides a gentle introduction to this popular subject. Assuming a background of a first course in abstract algebra, the book covers rings, ideals, quotients and homomorphisms, introduces polynomials, field extensions and splitting fields, gives a description of finite fields, and includes a brief account of the use of such fields in coding theory. It provides a readable, "student-friendly" introduction that takes a more "natural" approach to its subject (compared to the more formal introductions by Stewart and Garling), and that features clear explanations and plenty of worked examples and exercises - with full solutions to encourage independent study.

Modern and student-friendly introduction to this popular subject: it takes a more "natural" approach and develops the theory at a gentle pace with an emphasis on clear explanations Features plenty of worked examples and exercises, complete with full solutions, to encourage independent study Includes supplementary material: sn.pub/extras

Texte du rabat

The pioneering work of Abel and Galois in the early nineteenth century demonstrated that the long-standing quest for a solution of quintic equations by radicals was fruitless: no formula can be found. The techniques they used were, in the end, more important than the resolution of a somewhat esoteric problem, for they were the genesis of modern abstract algebra.

This book provides a gentle introduction to Galois theory suitable for third- and fourth-year undergraduates and beginning graduates. The approach is unashamedly unhistorical: it uses the language and techniques of abstract algebra to express complex arguments in contemporary terms. Thus the insolubility of the quintic by radicals is linked to the fact that the alternating group of degree 5 is simple - which is assuredly not the way Galois would have expressed the connection.

Topics covered include:

rings and fields

integral domains and polynomials

field extensions and splitting fields

applications to geometry

finite fields

the Galois group

equations

Group theory features in many of the arguments, and is fully explained in the text. Clear and careful explanations are backed up with worked examples and more than 100 exercises, for which full solutions are provided.

Contenu
Rings and Fields.- Integral Domains and Polynomials.- Field Extensions.- Applications to Geometry.- Splitting Fields.- Finite Fields.- The Galois Group.- Equations and Groups.- Some Group Theory.- Groups and Equations.- Regular Polygons.- Solutions.