|Original Title||Higher Algebra|
|Topics||mathematics, mir books, mir publishers, algebra, linear equations, matrices, polynomials, complex numbers, quadratic forms, linear spaces, fields, groups|
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Higher algebra—the subject of this text—is a far-reaching and natural generalization of the basic school course of elementary algebra. Central to elementary algebra is without doubt the problem of solving equations. The study of equations begins with the very simple case of one equation of the first degree in one unknown. From there on, the development proceeds in two directions: to systems of two and three equations of the first degree in two and, respectively, three unknowns, and to a single quadratic equation in one unknown and also to a few special types of higher-degree equations which readily reduce to quadratic equations (quartic equations, for example).
The second half of the course of higher algebra, called the algebra of polynomials, is devoted to the study of a single equation in one unknown but of arbitrary degree. Since there is a formula for solving quadratic equations, it was natural to seek similar formulas for higher-degree equations. That is precisely how this division of algebra developed historically. Formulas for solving equations of third and fourth degree were found in the sixteenth century. The search was then on for formulas capable of expressing the roots of equations of fifth and higher degree in terms of the coefficients of the equations by means of radicals, even radicals within radicals. It was futile, though it continued up to the beginning of the nine teenth century, when it was proved that no such formulas exist and that for all degrees beyond the fourth there even exist specific examples of equations with integral coefficients whose roots cannot be written down by means of radicals.