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, , are exactly zero.

For example, finding a substitution

for a cubic equation of degree ,
such that substituting yields a new equation
such 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.


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.[2][3]

Example: Tschirnhaus' method for cubic equationsEdit

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

using the Tschirnhaus transformation
Substituting yields the transformed equation
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.


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.


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  .[3]

See alsoEdit


  1. ^ a b von Tschirnhaus, Ehrenfried Walter; Green, R. F. (2003-03-01). "A method for removing all intermediate terms from a given equation". ACM SIGSAM Bulletin. 37 (1): 1–3. doi:10.1145/844076.844078. ISSN 0163-5824. S2CID 18911887.
  2. ^ Garver, Raymond (1927). "The Tschirnhaus Transformation". Annals of Mathematics. 29 (1/4): 319–333. doi:10.2307/1968002. ISSN 0003-486X. JSTOR 1968002.
  3. ^ a b Weisstein, Eric W. "Tschirnhausen Transformation". mathworld.wolfram.com. Retrieved 2022-02-02.