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.
For a generic degree
reduciblemonicpolynomial 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.[2][3]
Example: Tschirnhaus' method for cubic equations
In Tschirnhaus' 1683 paper,[1] he solved the equation
using the Tschirnhaus transformation
Substituting yields the transformed equation
or
Setting
yields,
and finally the Tschirnhaus transformation
which 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.
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
René Descartes to reduce a quadratic polynomial such that the 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 , , and for a general polynomial of degree .[3]
^
abC. B. Boyer (1968) A History of Mathematics. Wiley, New York pp. 472-473. As reported by: Weisstein, Eric W.
"Tschirnhausen Transformation". mathworld.wolfram.com. Retrieved 2022-02-02.