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. Simply, it is a method for transforming a polynomial equation of degree n ≥ 2 {\displaystyle n\geq 2} with some nonzero intermediate coefficients, a 1 , . . . , a n − 1 {\displaystyle a_{1},...,a_{n-1}} , such that some or all of the transformed intermediate coefficients, a 1 ′ , . . . , a n − 1 ′ {\displaystyle a'_{1},...,a'_{n-1}} , are exactly zero. For example, finding a substitution y ( x ) = k 1 x 2 + k 2 x + k 3 {\displaystyle y(x)=k_{1}x^{2}+k_{2}x+k_{3}} for a cubic equation of degree n = 3 {\displaystyle n=3} , f ( x ) = x 3 + a 2 x 2 + a 1 x + a 0 {\displaystyle f(x)=x^{3}+a_{2}x^{2}+a_{1}x+a_{0}} such that substituting x = x ( y ) {\displaystyle x=x(y)} yields a new equation f ′ ( y ) = y 3 + a 2 ′ y 2 + a 1 ′ y + a 0 ′ {\displaystyle f'(y)=y^{3}+a'_{2}y^{2}+a'_{1}y+a'_{0}} such that a 1 ′ = 0 {\displaystyle a'_{1}=0} , a 2 ′ = 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.
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