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. 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.
In detail, let formula_1 be a field, and formula_2 a polynomial over formula_1. If formula_4 is irreducible, then the quotient ring of the polynomial ring formula_5 by the principal ideal generated by formula_4,
is a field extension of formula_1. We have
where formula_10 is formula_11 modulo formula_12. That is, any element of formula_13 is a polynomial in formula_10, which is thus a primitive element of formula_13. There will be other choices formula_16 of primitive element in formula_13: for any such choice of formula_16 we will have by definition:
with polynomials formula_20 and formula_21 over formula_1. Now if formula_23 is the minimal polynomial for formula_16 over formula_1, we can call formula_23 a Tschirnhaus transformation of formula_4.
Therefore the set of all Tschirnhaus transformations of an irreducible polynomial is to be described as running over all ways of changing formula_4, but leaving formula_13 the same. This concept is used in reducing quintics to Bring–Jerrard form, for example. There is a connection with Galois theory, when formula_13 is a Galois extension of formula_1. The Galois group may then be considered as all the Tschirnhaus transformations of formula_4 to itself.