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''Tschirnhaus transformation''

A transformation of an $n$th degree polynomial equation

$$f(X)=X^n+a_{n-1}X^{n-1}+\ldots+a_0=0$$

by a substitution of the form

$$Y=\alpha_0+\alpha_1X+\ldots+\alpha_{n-1}X^{n-1}$$

to an equation

$$Y^n+b_{n-1}Y^{n-1}+\ldots+b_0=0,$$

hopefully of simpler form. For instance, the general equation of degree five can be brought to the form $Y^5+Y+b=0$ (the so-called Bring–Jerrard normal form) using only quadratic roots. Quite generally, the terms of degree $n-1$, $n-2$, $n-3$ can always be eliminated by a suitable Tschirnhausen transformation.

The procedure is named after Count E.W. von Tschirnhaus, who described these transformations in [[#References|[a1]]]. Contrary to the beliefs of Tschirnhaus and Jerrard at that time (around 1683), these transformations do not help solving general polynomial equations of degree larger than four (see also [[Galois theory|Galois theory]]).

A generalization of the Tschirnhausen transformation plays a role in the original proof of the [[Abhyankar–Moh theorem|Abhyankar–Moh theorem]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.W. von Tschirnhaus,   ''Acta Eruditorium''  (1683)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Weber,   "Lehrbuch der Algebra" , '''I''' , Chelsea, reprint  pp. Chap. 6  (First ed.: 1898)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.L. Cayley,   "On Tschirnhausen's transformation"  ''Philos. Trans. R. Soc. London'' , '''151'''  (1861)  pp. 561–578</TD></TR></table>