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Galois Theory and Its Algebraic Background

2nd Edition

textbook
  • Date Published: July 2021
  • availability: Available
  • format: Paperback
  • isbn: 9781108969086

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  • Galois Theory, the theory of polynomial equations and their solutions, is one of the most fascinating and beautiful subjects of pure mathematics. Using group theory and field theory, it provides a complete answer to the problem of the solubility of polynomial equations by radicals: that is, determining when and how a polynomial equation can be solved by repeatedly extracting roots using elementary algebraic operations. This textbook contains a fully detailed account of Galois Theory and the algebra that it needs and is suitable both for those following a course of lectures and the independent reader (who is assumed to have no previous knowledge of Galois Theory). The second edition has been significantly revised and re-ordered; the first part develops the basic algebra that is needed, and the second a comprehensive account of Galois Theory. There are applications to ruler-and- compass constructions, and to the solution of classical mathematical problems of ancient times. There are new exercises throughout, and carefully-selected examples will help the reader develop a clear understanding of the mathematical theory.

    • The revised second edition, with more examples and additional background material
    • Contains a wealth of new exercises to challenge the reader
    • Gives a direct and straightforward account of the mathematical theory
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    Reviews & endorsements

    'Garling's book presents Galois theory in a style which is at once readable and compact. The necessary prerequisites are developed in the early chapters only to the extent that they are needed later. The proofs of the lemmas and main theorems are presented in as concrete a manner as possible, without unnecessary abstraction. Yet they seem remarkably short, without the difficulties being glossed over. In fact the approach throughout the book is down-to-earth and concrete … I can heartily recommend this book as an undergraduate text.' Bulletin of the London Mathematical Society

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    Product details

    • Edition: 2nd Edition
    • Date Published: July 2021
    • format: Paperback
    • isbn: 9781108969086
    • length: 204 pages
    • dimensions: 229 x 151 x 18 mm
    • weight: 0.36kg
    • availability: Available
  • Table of Contents

    Part I. The Algebraic Background:
    1. Groups
    2. Integral domains
    3. Vector spaces and determinants
    Part II. The Theory of Fields, and Galois Theory:
    4. Field extensions
    5. Ruler and compass constructions
    6. Splitting fields
    7. Normal extensions
    8. Separability
    9. The fundamental theorem of Galois theory
    10. The discriminant
    11. Cyclotomic polynomials and cyclic extensions
    12. Solution by radicals
    13. Regular polygons
    14. Polynomials of low degree
    15. Finite fields
    16. Quintic polynomials
    17. Further theory
    18. The algebraic closure of a field
    19. Transcendental elements and algebraic independence
    20. Generic and symmetric polynomials
    Appendix: the axiom of choice
    Index.

  • Author

    D. J. H. Garling, University of Cambridge
    D. J. H. Garling is Emeritus Reader in Mathematical Analysis at the University of Cambridge and Fellow of St John's College, Cambridge. He has fifty years' experience of teaching undergraduate students and has written several books on mathematics, including Inequalities: A Journey into Linear Analysis (Cambridge, 2007) and A Course in Mathematical Analysis (Three volumes, Cambridge, 2013–2014).

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