Galois' Theory of Algebraic EquationsWorld Scientific, 2001 - 333 lappuses Galois' Theory of Algebraic Equations gives a detailed account of the development of the theory of algebraic equations, from its origins in ancient times to its completion by Galois in the nineteenth century. The main emphasis is placed on equations of at least the third degree, i.e. on the developments during the period from the sixteenth to the nineteenth century. The appropriate parts of works by Cardano, Lagrange, Vandermonde, Gauss, Abel and Galois are reviewed and placed in their historical perspective, with the aim of conveying to the reader a sense of the way in which the theory of algebraic equations has evolved and has led to such basic mathematical notions as ?group? and ?field?. A brief discussion on the fundamental theorems of modern Galois theory is included. Complete proofs of the quoted results are provided, but the material has been organized in such a way that the most technical details can be skipped by readers who are interested primarily in a broad survey of the theory.This book will appeal to both undergraduate and graduate students in mathematics and the history of science, and also to teachers and mathematicians who wish to obtain a historical perspective of the field. The text has been designed to be self-contained, but some familiarity with basic mathematical structures and with some elementary notions of linear algebra is desirable for a good understanding of the technical discussions in the later chapters. |
Saturs
Preface | 1 |
Cubic Equations | 13 |
Quartic Equations | 21 |
A Modern Approach to Polynomials | 41 |
Decomposition of rational fractions in sums of partial fractions | 58 |
4 | 67 |
1 | 73 |
Leibniz and Newton on the summation of series | 92 |
Vandermonde | 153 |
Gauss on Cyclotomic Equations | 167 |
25 | 169 |
Ruler and compass construction of regular polygons | 200 |
Exercises | 227 |
41 | 291 |
Galois description of groups of permutations | 295 |
Exercises | 315 |
Eulers summation of the series of reciprocals of perfect squares | 110 |
Lagrange | 123 |
Exercises | 150 |
331 | |
Citi izdevumi - Skatīt visu
Galois' Theory of Algebraic Equations: Second Edition Jean-Pierre Tignol Ierobežota priekšskatīšana - 2015 |
Bieži izmantoti vārdi un frāzes
a₁ algebra assume b₁ base field calculations Cardano's Cardano's formula complex numbers containing F Corollary cosets cubic equation cyclotomic equations denote divides divisor element elementary symmetric polynomials equation of degree exponent expression by radicals extension of F extension of height field containing field F finite formula function fundamental theorem GA(p Gal(P/F Galois group Galois resolvent Gauss hence indeterminates integer invariant irreducible polynomial Lagrange Lagrange's Lemma linear factors method minimum polynomial modulo Moivre's monic polynomial Moreover non-zero notation obtained p-th power P₁ pairwise distinct permutations of x1 polynomial of degree prime index prime number primitive n-th root product of linear Proposition prove quadratic equation r₁ radical extension rational expressions rational fraction readily follows relatively prime result roots of unity s₁ shows solution solvable by radicals solving subfield substituting u₁ V₁ values Vandermonde whence yields
Atsauces uz šo grāmatu
Mathematical Expeditions: Chronicles by the Explorers Reinhard Laubenbacher,David Pengelley Ierobežota priekšskatīšana - 2000 |