Fields and Galois Theory (Springer Undergraduate Mathematics Series)
Summary:
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.
Synopsis:
Aimed at 3rd and 4th year undergraduates and beginning graduates, this book provides a gentle introduction to this popular subject. Assuming a background of a first course in abstract algebra, the book begins with a review of rings, ideals, quotients and homomorphisms. Polynomials, a key topic in field theory, are then introduced in the second chapter. Field extensions and splitting fields are the topics of Chapters 3 and 4, and there is an account of ruler and compass constructions, and a proof that squaring the circle is impossible, in Chapter 5. Chapter 6 uses the theory developed in Chapters 3 and 4 to give a description of finite fields, and includes a brief account of the use of such fields in coding theory. The book then concludes with the Galois group, normal and separable extensions, an account of polynomial equations, and the celebrated result that the quintic equation is not soluble by radicals. The aim is to provide a readable, student-friendly introduction that takes a more natural approach to its subject (as 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
http://rapidshare.com/files/12554779/Fields.and.Galois.Theory-1852339861.pdf
http://ifile.it/w1nikf/fields.and.galois.theory-1852339861.pdf
