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Galois Theory

BuchKartoniert, Paperback
Englisch
Springer Netherlanderschienen am03.10.2011Softcover reprint of the original 1st ed. 1989
Galois theory is a showpiece of mathematical unification, bringing together several different branches of the subject and creating a power- ful machine for the study of problems of considerable historical and mathematical importance. This book is an attempt to present the theory in such a light, and in a manner suitable for second- and third-year undergraduates. The central theme is the application of the Galois group to the quintic equation. As well as the traditional approach by way of the 'general' polynomial equation I have included a direct approach which demon- strates the insolubility by radicals of a specific quintic polynomial with integer coefficients, which I feel is a more convincing result. The abstract Galois theory is set in the context of arbitrary field extensions, rather than just subfields of the complex numbers; the resulting gain in generality more than compensates for the extra work required. Other topics covered are the problems of duplicating the cube, trisecting the angle, and squaring the circle; the construction of regular polygons; the solution of cubic and quartic equations; the structure of finite fields; and the 'fundamental theorem of algbra'.The last is proved by almost purely algebraic methods, and provides an interesting application of Sylow theory. In order to make the treatment as self-contained as possible, and to bring together all the relevant material in a single volume, I have included several digressions.mehr
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BuchKartoniert, Paperback
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TaschenbuchKartoniert, Paperback
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Produkt

KlappentextGalois theory is a showpiece of mathematical unification, bringing together several different branches of the subject and creating a power- ful machine for the study of problems of considerable historical and mathematical importance. This book is an attempt to present the theory in such a light, and in a manner suitable for second- and third-year undergraduates. The central theme is the application of the Galois group to the quintic equation. As well as the traditional approach by way of the 'general' polynomial equation I have included a direct approach which demon- strates the insolubility by radicals of a specific quintic polynomial with integer coefficients, which I feel is a more convincing result. The abstract Galois theory is set in the context of arbitrary field extensions, rather than just subfields of the complex numbers; the resulting gain in generality more than compensates for the extra work required. Other topics covered are the problems of duplicating the cube, trisecting the angle, and squaring the circle; the construction of regular polygons; the solution of cubic and quartic equations; the structure of finite fields; and the 'fundamental theorem of algbra'.The last is proved by almost purely algebraic methods, and provides an interesting application of Sylow theory. In order to make the treatment as self-contained as possible, and to bring together all the relevant material in a single volume, I have included several digressions.
Details
ISBN/GTIN978-94-010-6864-2
ProduktartBuch
EinbandartKartoniert, Paperback
ErscheinungsortDordrecht
ErscheinungslandNiederlande
Erscheinungsjahr2011
Erscheinungsdatum03.10.2011
AuflageSoftcover reprint of the original 1st ed. 1989
SpracheEnglisch
Gewicht367 g
Artikel-Nr.29947211

Inhalt/Kritik

Inhaltsverzeichnis
1 Background.- 2 Factorization of polynomials.- 3 Field extensions.- 4 The degree of an extension.- 5 Ruler and compasses.- 6 Transcendental numbers.- 7 The idea behind Galois theory.- 8 Normality and separability.- 9 Field degrees and group orders.- 10 Monomorphisms, automorphisms, and normal closures.- 11 The Galois correspondence.- 12 A specific example.- 13 Soluble and simple groups.- 14 Solution of equations by radicals.- 15 The general polynomial equation.- 16 Finite fields.- 17 Regular polygons.- 18 Calculating Galois groups.- 19 The 'fundamental theorem of algebra'.- Selected solutions.- References.- Symbol Index.mehr