By A.G. Kurosh, V. Kisin

This e-book is a revision of the author's lecture to school scholars playing the math Olympiad at Moscow kingdom college. It provides a overview of the implications 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, identified for his paintings in summary algebra. he's credited with writing the 1st glossy and high-level textual content on team conception, "The thought of Groups", released in 1944. CONTENTS: Preface / advent / 1. advanced Numbers 2. Evolution. Quadratic Equations three. Cubic Equations four. answer of Equations by way of Radicals and the lifestyles of Roots of Equations five. The variety of genuine Roots 6. Approximate answer of Equations 7. Fields eight. end / Bibliography

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In addition elementary algebra proceeds from a study of one first-degree equation with one variable to a system of two first-degree equations with two variables and a system of three equations with three variables. A university course in higher algebra continues these trends and teaches the methods for solving -any system of n first-degree equations with n variables, and also the methods of solving such systems of first-degree equations in which the number of equations is not equal to the number of variables.

Besides these three, an infinite number of other fields exist. For instance, many different fields are contained within the fields of real numbers and of complex numbers; these are the so-called numerical fields. In addition some fields are larger than that of complex numbers. The elements of these fields are no longer called numbers, but the fields formed by them are used in mathematical research. Here is one example of such a field. Let us consider all possible polynomials f( x ) = aoxn + alx n-l + ...

E. a + Oa = a (1) If b is any other element of P, then again there exists one such unique element O, for which (2) If we prove that Oa = O, for any a and b, then the existence in the set P of an element, which plays the role of zero for all the elements a at the same time, will be immediately proved. Let c be the root of the equation a+x=b which exists because of Condition IV; hence, a+c=b We now add to both sides of equation (1) the element c, which does not violate the equality due to the uniqueness of the sum: (a + 0a) +c= a+c The right-hand side of this equation equals b, and the left-hand side, due to Conditions I and II, equals b + Oa.

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