By A.G. Kurosh, V. Kisin

This book is a revision of the author's lecture to school scholars enjoying the maths Olympiad at Moscow kingdom collage. It provides a overview of the consequences and strategies of the overall conception of algebraic equations with due regard for the extent of data of its readers. Aleksandr Gennadievich Kurosh (1908-1971) was once a Soviet mathematician, identified for his paintings in summary algebra. he's credited with writing the 1st sleek and high-level textual content on crew concept, "The conception of Groups", released in 1944. CONTENTS: Preface / advent / 1. complicated Numbers 2. Evolution. Quadratic Equations three. Cubic Equations four. answer of Equations by way of Radicals and the life of Roots of Equations five. The variety of actual Roots 6. Approximate answer of Equations 7. Fields eight. end / Bibliography

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Lobachevsky (1793-1856), the creator of non-Euclidean geometry. 7. Fields The problem of roots of algebraic equations, which we have already encountered above, can be considered in more general terms. To do so we must introduce one of the most important concepts of algebra. Let us first consider the following three systems of numbers: the set of all rational numbers, the set of all real numbers, and the set of all complex numbers. Without leaving their respective bounds, we can add, multiply, subtract and divide (except for division by zero) in each of these systems of numbers.

E. for any a and b a + b = b + a, ab = ba II. Both operations are associative, i. e. for any a, band c (a + b) + c = a + (b + c), (ab) c = a (bc) III. The law of distribution of multiplication with respect to addition holds, i. e. for any a, band c a (b + c) = ab + ac IV. Subtraction can be carried out, i. e. for any a and b a unique root of the equation a+x=b can be found in P. V. Division can be carried out, i. e. for any a and b, provided a does not equal zero, a unique root of the equation ax = b can be found in P.

S. Aleksandrov, Introduction to Group Theory, "Uchpedgiz", 1951 (in Russian). TO THE READER Mir Publishers would be grateful for your comments on the content, translation and design of this book. We would also be pleased to receive any other suggestions you may wish to make. Our address is: USSR, 129820, Moscow 1-110, GSP Pervy Rizhsky Pereulok, 2 Mir Publishers Printed in the Union of Soviet Socialist Republics Other Books by MIR PUBLISHERS from LfITLE MATHEMATICS LIBRARY Series G. E. Shilov CALCULUS OF RATIONAL FUNCTIONS An introduction to the principal concepts of mathematical analysis (the derivative and the integral) within the comparatively limited field of rational functions, employing the visual language of graphs.