АЛГЕБРАИЧЕСКИЕ УРАВНЕНИЯ В УНИТАРНОМ ПРОСТРАНСТВЕ И КРАТЧАЙШЕЕ АЛГЕБРАИЧЕСКОЕ ДОКАЗАТЕЛЬСТВО ОСНОВНОЙ ТЕОРЕМЫ АЛГЕБРЫ

Abstract

The article deals with the presentation of some results obtained in the study of algebraic equations with complex coefficients from a complex variable in a unitary space. In the orthonormal basis, two vectors are introduced, which are called the vector of root and the vector of coefficients of an algebraic polynomial. With the help of these vectors, an algebraic polynomial is represented as a scalar product of them in an orthonormal basis. The criterion of linear independence of a set of root vectors is formulated and proved. A Theorem is formulated and proved that the maximum number of simple roots of an algebraic polynomial is one less than the dimension of a unitary space. A generalization of Vieta’s Theorem is obtained and new formulas connecting the coefficients of an algebraic polynomial with its roots are derived. The general case of an algebraic polynomial is considered, when some of its coefficients may be equal to zero. Two orthogonal vector systems from combinations of coefficients of an algebraic polynomial are introduced. Their properties have been studied. The shortest algebraic proof of the fundamental Theorem of algebra among the known proofs is given, which does not go beyond the concepts of the algebra of polynomials and uses the scalar product of vectors in a unitary space, as well as one property of a two-place predicate from mathematical logic. The results of this work can also be used in the educational process in the algebra course.