Tschirnhaus transformation

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.
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)