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 ${\displaystyle K}$ be a field, and ${\displaystyle P(t)}$ a polynomial over ${\displaystyle K}$. If ${\displaystyle P}$ is irreducible, then the quotient ring of the polynomial ring ${\displaystyle K[t]}$ by the principal ideal generated by ${\displaystyle P}$,

${\displaystyle K[t]/(P(t))=L}$,

is a field extension of ${\displaystyle K}$. We have

${\displaystyle L=K(\alpha )}$

where ${\displaystyle \alpha }$ is ${\displaystyle t}$ modulo ${\displaystyle (P)}$. That is, any element of ${\displaystyle L}$ is a polynomial in ${\displaystyle \alpha }$, which is thus a primitive element of ${\displaystyle L}$. There will be other choices ${\displaystyle \beta }$ of primitive element in ${\displaystyle L}$: for any such choice of ${\displaystyle \beta }$ we will have by definition:

${\displaystyle \beta =F(\alpha ),\alpha =G(\beta )}$,

with polynomials ${\displaystyle F}$ and ${\displaystyle G}$ over ${\displaystyle K}$. Now if ${\displaystyle Q}$ is the minimal polynomial for ${\displaystyle \beta }$ over ${\displaystyle K}$, we can call ${\displaystyle Q}$ a Tschirnhaus transformation of ${\displaystyle P}$.

Therefore the set of all Tschirnhaus transformations of an irreducible polynomial is to be described as running over all ways of changing ${\displaystyle P}$, but leaving ${\displaystyle L}$ the same. This concept is used in reducing quintics to Bring–Jerrard form, for example. There is a connection with Galois theory, when ${\displaystyle L}$ is a Galois extension of ${\displaystyle K}$. The Galois group may then be considered as all the Tschirnhaus transformations of ${\displaystyle P}$ to itself.

See also

• Polynomial transformations

References

• Weisstein, Eric W. "Tschirnhausen Transformation". MathWorld.
• Tschirnhaus's 1683 paper "A method for removing all intermediate terms from a given equation", translation by RF Green (2003).