INTEGRATED MODERN METHODOLOGY FOR RATIONALIZING COMPLEX IRRATIONAL DENOMINATORS IN HIGHER ALGEBRA AND SCHOOL MATHEMATICS
Keywords:
Irrationality, conjugate expression, algebraic extensions, minimal polynomial, extended Euclidean algorithm, rationalizing factor.Abstract
This article is devoted to a modern integrated methodology for teaching the problem of rationalizing denominators of fractions containing complex irrational expressions — particularly in cases involving sums or products of several cube roots with distinct bases — within school mathematics and higher algebra curricula. It is demonstrated that traditional methods (conjugate expressions, rationalizing conjugates) are limited to single square roots or simple cubic roots. For more intricate cases, solutions are proposed based on fundamental tools of algebraic number theory: the minimal polynomial, the basis of an algebraic number field extension, and the computation of the inverse element within that extension.
References
1. R.N.Nazarov, B.T.Toshpo‘latov, A.D.Do‘sumbetov. Algebra va sonlar nazariyasi. II qism.O‘quv qo‘llanma. Toshkent. O‘qituvchi. 1995 yil.
2. Lang, S. Algebra. Springer-Verlag. (2002).
3. Childs, L. N. A Concrete Introduction to Higher Algebra. Springer. (2009).
4. Shoup, V. A Computational Introduction to Number Theory and Algebra. Cambridge University Press. (2009).
