By A.G. Kurosh, V. Kisin

This ebook is a revision of the author's lecture to highschool scholars enjoying the math Olympiad at Moscow nation collage. It offers a evaluate of the consequences and techniques of the final thought 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 glossy and high-level textual content on workforce thought, "The idea of Groups", released in 1944. CONTENTS: Preface / creation / 1. complicated Numbers 2. Evolution. Quadratic Equations three. Cubic Equations four. answer of Equations when it comes to Radicals and the lifestyles of Roots of Equations five. The variety of actual Roots 6. Approximate answer of Equations 7. Fields eight. end / Bibliography

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The basic theorem of higher algebra shows that the field of complex numbers belongs to the set of algebraically closed fields. 8. Conclusion Throughout this booklet we always discussed equations of a certain degree with one variable. The study of first-degree equations is followed by that of quadratic equations in elementary algebra. 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.

Let us consider all possible polynomials f( x ) = aoxn + alx n-l + ... + an-IX + an with arbitrary complex coefficients and of arbitrary degrees; for instance, zero-degree polynomials will be represented by complex numbers themselves. Even if we add, subtract and multiply polynomials with complex coefficients by the rules we already know, we still will not obtain a· field, since division of a polynomial by another polynomial with no remainder is not always possible. Now let us consider ratios of polynomials f(x) g(x) or rational functions with complex coefficients, and let us agree to treat them in the way we treat fractions.

Proskuryakov, Numbers and Polynomials, "Prosveshchenie", 1965 (in Russian). An acquaintance with group· theory may begin with: 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.

### Algebraic Equations of Arbitrary Degrees by A.G. Kurosh, V. Kisin

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