In mathematics, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomials developed by Ehrenfried Walther von Tschirnhaus in 1683.[1]
Simply, it is a method for transforming a polynomial equation of degree with some nonzero intermediate coefficients, , such that some or all of the transformed intermediate coefficients, , are exactly zero.
For example, finding a substitutionfor a cubic equation of degree ,such that substituting yields a new equationsuch that , , or both.
More generally, it may be defined conveniently by means of field theory, as the transformation on minimal polynomials implied by a different choice of primitive element. This is the most general transformation of an irreducible polynomial that takes a root to some rational function applied to that root.