Tschirnhaus transformation

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 ${\displaystyle n\geq 2}$ with some nonzero intermediate coefficients, ${\displaystyle a_{1},...,a_{n-1}}$, such that some or all of the transformed intermediate coefficients, ${\displaystyle a'_{1},...,a'_{n-1}}$, are exactly zero.

For example, finding a substitution$${\displaystyle y(x)=k_{1}x^{2}+k_{2}x+k_{3}}$$for a cubic equation of degree ${\displaystyle n=3}$,$${\displaystyle f(x)=x^{3}+a_{2}x^{2}+a_{1}x+a_{0}}$$such that substituting ${\displaystyle x=x(y)}$ yields a new equation$${\displaystyle f'(y)=y^{3}+a'_{2}y^{2}+a'_{1}y+a'_{0}}$$such that ${\displaystyle a'_{1}=0}$, ${\displaystyle a'_{2}=0}$, 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.