Buch, Englisch, PB, Format (B × H): 155 mm x 235 mm, Gewicht: 367 g
Buch, Englisch, PB, Format (B × H): 155 mm x 235 mm, Gewicht: 367 g
ISBN: 978-94-010-6864-2
Verlag: Springer Netherlands
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.
Zielgruppe
Research
Autoren/Hrsg.
Weitere Infos & Material
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.
