By A.G. Kurosh, V. Kisin

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

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E. u(x) # 0, f(x) u (x) f(x) v(x) g(x) : v(x) = g(x)u(x) It can easily be checked that the above operations with rational functions satisfy all the requirements of the definition of a field, so that we can speak of a field of rational functions with complex coefficients. The field of complex numbers is totally contained in this field, since a rational function whose numerator and denominator are zero-degree polynomials is simply a complex number, and any complex number can be presented in this form.

E. ones in which the product is changed by commutation of co-factors, and sometimes nonassociatioe operations, i. e. those in which the product of three factors depends on the location of brackets. Those groups which are used to solve equations in radicals are noncommutative. 35 Systematic presentation of the fundamentals of the theory of algebraic equations and of linear algebra can be found in textbooks on higher algebra. The following textbooks are most frequently recommended: A. G. Kurosh, Higher Algebra, Mir Publishers, 1975 (in English).

2an - 2 X + £In - 1 - 2) U2 X"- 3 + ... a derivative. of this polynomial, and denote it as f'(x). This polynomial is derived from f(x) by the following rule: each term akxn-k of the polynomial f(x) is multiplied by the exponent n. - k of x, while the exponent itself is reduced by unity; moreover, the absolute term an disappears, since we can consider that an = anxo. We can again take the derivative of the polynomial f' (x). This will be a polynomial of degree (n - 2), which is called the second derivative of the polynomial f(x) and is denoted as f" (x).