Tschirnhaus transformations

A polynomial transformation which transforms a polynomialMathworldPlanetmathPlanetmathPlanetmath to another with certain zero-coefficients is called a Tschirnhaus Transformation. It is thus an invertible transformation of the form xg(x)/h(x) where g,h are polynomials over the base fieldMathworldPlanetmathPlanetmath K (or some subfieldMathworldPlanetmath of the splitting fieldMathworldPlanetmath of the polynomial being transformed). If gcd(h(x),f(x))=1 then the Tschirnhaus transformation becomes a polynomial transformation mod f.

Specifically, it concerns a substitution that reduces finding the roots of the polynomial


to finding the roots of another q - with less parameters - and solving an auxiliary polynomial equation s, with deg(s)<deg(pq).

Historically, the transformation was applied to reduce the general quintic equation, to simpler resolvents. Examples due to Hermite and Klein are respectively: The principal resolvent


and the Bring-Jerrard form


Tschirnhaus transformations are also used when computing Galois groups to remove repeated roots in resolvent polynomials. Almost any transformation will work but it is extremely hard to find an efficient algorithm that can be proved to work.

Title Tschirnhaus transformations
Canonical name TschirnhausTransformations
Date of creation 2013-03-22 13:50:12
Last modified on 2013-03-22 13:50:12
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 11
Author mathcam (2727)
Entry type Definition
Classification msc 12E05
Synonym Tschirnhausen Transformation
Related topic QuadraticResolvent
Related topic EulersDerivationOfTheQuarticFormula