By A.G. Kurosh, V. Kisin

This publication is a revision of the author's lecture to highschool scholars playing the maths Olympiad at Moscow country college. It provides a assessment of the consequences and strategies of the overall concept of algebraic equations with due regard for the extent of information of its readers. Aleksandr Gennadievich Kurosh (1908-1971) was once a Soviet mathematician, recognized for his paintings in summary algebra. he's credited with writing the 1st sleek and high-level textual content on team conception, "The thought of Groups", released in 1944. CONTENTS: Preface / creation / 1. advanced Numbers 2. Evolution. Quadratic Equations three. Cubic Equations four. answer of Equations when it comes to Radicals and the life of Roots of Equations five. The variety of genuine Roots 6. Approximate resolution of Equations 7. Fields eight. end / Bibliography

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**Example text**

0215 ... 7784 ... 7769 ... 7784 ... 7785 It follows, therefore, that if we take for (X2 the arithmetic mean, i. e. 0008, equal to half the difference of these bounds. If the resulting accuracy is insufficient, we could once again apply the above method to the new bounds of the root (X2. However, this would require much more complicated calculations. Other methods of approximate solution of equations are more accurate. The best method, that permits the approximate calculation of not only the real but also the complex roots of equations, was devised by the great Russian mathematician N.

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.

However, this would require much more complicated calculations. Other methods of approximate solution of equations are more accurate. The best method, that permits the approximate calculation of not only the real but also the complex roots of equations, was devised by the great Russian mathematician N. I. 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.