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Fields and Galois Theory
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103 b
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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.
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