In mathematics, a Tschirnhaus transformation, developed by Ehrenfried Walther von Tschirnhaus in 1683, 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. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Ehrenfried Walther von Tschirnhaus (or Tschirnhausen) (April 10, 1651–October 11, 1708) was a German mathematician. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... Field theory is a branch of mathematics which studies the properties of fields. ... In mathematics, the minimal polynomial of an object α is the monic polynomial p of least degree such that p(α)=0. ... In mathematics, a primitive element for an extension of fields L/K is an element ζ of L such that L = K(ζ), or in other words such that L is generated by ζ over K. This means that every element of L can be written as a quotient of two polynomials in... In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ...

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 In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In Ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R. More specifically: a left principal ideal of R is a subset of R of the form Ra := {ra : r in R... In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. ...

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. In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. ... In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions (described below); one also says that the extension is Galois. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ...

References

• M. Hazewinkel (2001), "Tschirnhausen transformation", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
• Eric W. Weisstein, Tschirnhausen Transformation at MathWorld.