#jsDisabledContent { display:none; } My Account | Register | Help

# Tschirnhaus transformation

Article Id: WHEBN0000595824
Reproduction Date:

 Title: Tschirnhaus transformation Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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 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.

## References

• Weisstein, Eric W., "Tschirnhausen Transformation", MathWorld.
• http://www.sigsam.org/bulletin/articles/143/tschirnhaus.pdf A translation (by RF Green) of his 1683 paper—A method for removing all intermediate terms from a given equation.

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.