# Tschirnhaus transformation

In mathematics, a **Tschirnhaus transformation** is a type of mapping on polynomials. 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 *K* be a field, and *P*(*t*) a polynomial over *K*. If *P* is irreducible, then

*K*[*t*]/(*P*(*t*)) =*L*,

the quotient ring of the polynomial ring *K*[*t*] by the principal ideal generated by *P*, is a field extension of *K*. We have

*L*=*K*(α)

where α is *t* modulo (*P*). That is, α is a primitive element of *L*. There will be other choices β of primitive element in *L*: for any such choice of β we will have

- β =
*F*(α), α =*G*(β),

with polynomials *F* and *G* over *K*. In fact this follows from the quotient representation above. Now if *Q* is the minimal polynomial for β over *K*, we can call *Q* a **Tschirnhaus transformation** of *P*.

Therefore the set of all Tschirnhaus transformations of an irreducible polynomial is to be described as running over all ways of changing *P*, but leaving *L* the same. This concept is used in reducing quintics to Bring-Jerrard form, for example. There is a connection with Galois theory, when *L* is a Galois extension of *K*. The Galois group is then described (in one way) as all the Tschirnhaus transformations of *P* to itself.