Fields and Galois Theory (eBook)

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.

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Fieldsaresetsinwhichallfouroftherationaloperations,memorablydescribed by the mathematician Lewis Carroll as "e;perdition, distraction, ugli?cation and derision"e;, can be carried out. They are assuredly the most natural of algebraic objects, since most of mathematics takes place in one ?eld or another, usually the rational ?eld Q, or the real ?eld R, or the complex ?eld C. This book sets out to exhibit the ways in which a systematic study of ?elds, while interesting in its own right, also throws light on several aspects of classical mathematics, notably on ancient geometrical problems such as "e;squaring the circle"e;, and on the solution of polynomial equations. The treatment is unashamedly unhistorical. When Galois and Abel dem- strated that a solution by radicals of a quintic equation is not possible, they dealt with permutations of roots. From sets of permutations closed under c- position came the idea of a permutation group, and only later the idea of an abstract group. In solving a long-standing problem of classical algebra, they laid the foundations of modern abstract algebra.