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  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.

Definition 
For a generic  degree reducible monic polynomial equation  of the form , where  and  are polynomials and  does not vanish at ,the Tschirnhaus transformation is the function:Such that the new equation in , , has certain special properties, most commonly such that some coefficients, , are identically zero.

Example: Tschirnhaus' method for cubic equations 
In Tschirnhaus' 1683 paper, he solved the equationusing the Tschirnhaus transformationSubstituting yields the transformed equationorSetting  yields,and finally the Tschirnhaus transformationWhich may be substituted into  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.

Generalization 
In detail, let  be a field, and  a polynomial over . If  is irreducible, then the quotient ring of the polynomial ring  by the principal ideal generated by ,

,

is a field extension of . We have

where  is  modulo . That is, any element of  is a polynomial in , which is thus a primitive element of . There will be other choices  of primitive element in : for any such choice of  we will have by definition:

,

with polynomials  and  over . Now if  is the minimal polynomial for  over , we can call  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 , but leaving  the same. 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  to itself.

History 
In 1683, Ehrenfried Walther von Tschirnhaus published a method for rewriting a polynomial of degree  such that the  and  terms have zero coefficients. In his paper, Tschirnhaus referenced a method by Descartes to reduce a quadratic polynomial  such that the  term has zero coefficient.

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

In 1834, G. B. Jerrard further expanded Tschirnhaus' work by showing a Tschirnhaus transformation may be used to eliminate the , , and  for a general polynomial of degree .

See also
Polynomial transformations
Monic polynomial
Reducible polynomial
Quintic function
Galois theory
Abel-Ruffini theorem

References

Polynomials
Field (mathematics)