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

For example, finding a substitution

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.

History[edit]

In 1683, Ehrenfried Walther von Tschirnhaus published a method for rewriting a polynomial of degree $n>2$ such that the $x^{n-1}$ and $x^{n-2}$ terms have zero coefficients. In his paper, Tschirnhaus referenced a method by René Descartes to reduce a quadratic polynomial $(n=2)$ such that the $x$ term has zero coefficient.

In 1786, this work was expanded by Erland Samuel Bring who showed that any generic quintic polynomial could be similarly reduced.

In 1834, George Jerrard further expanded Tschirnhaus' work by showing a Tschirnhaus transformation may be used to eliminate the $x^{n-1}$ , $x^{n-2}$ , and $x^{n-3}$ for a general polynomial of degree $n>3$ .^[3]

Tschirnhaus transformation

History[edit]

Polynomial transformations

Monic polynomial

Reducible polynomial

Quintic function

Galois theory

Abel-Ruffini theorem