Tschirnhaus transformation

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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.

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