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

with some nonzero intermediate coefficients,

, such that some or all of the transformed intermediate coefficients,

for a cubic equation of degree

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.

degree reducible monic polynomial equation

the Tschirnhaus transformation is the function:

, has certain special properties, most commonly such that some coefficients,

[2][3] In Tschirnhaus' 1683 paper,[1] he solved the equation

using the Tschirnhaus transformation

Substituting yields the transformed equation

and finally the Tschirnhaus transformation

to yield an equation of the form:

Tschirnhaus went on to describe how a Tschirnhaus transformation of the form:

may be used to eliminate two coefficients in a similar way.

is irreducible, then the quotient ring of the polynomial ring

by the principal ideal generated by

a Tschirnhaus transformation of

Therefore the set of all Tschirnhaus transformations of an irreducible polynomial is to be described as running over all ways of changing

This concept is used in reducing quintics to Bring–Jerrard form, for example.

There is a connection with Galois theory, when

is a Galois extension of

The Galois group may then be considered as all the Tschirnhaus transformations of

In 1683, Ehrenfried Walther von Tschirnhaus published a method for rewriting a polynomial of degree

In his paper, Tschirnhaus referenced a method by René Descartes to reduce a quadratic polynomial

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

for a general polynomial of degree