Tschirnhausen Transformation

A transformation of a Polynomial equation $f(x)=0$ which is of the form $y=g(x)/h(x)$ where $g$ and $h$ are Polynomials and $h(x)$ does not vanish at a root of $f(x)=0$. The Cubic Equation is a special case of such a transformation. Tschirnhaus (1683) showed that a Polynomial of degree $n>2$ can be reduced to a form in which the $x^{n-1}$ and $x^{n-2}$ terms have 0 Coefficients. In 1786, E. S. Bring showed that a general Quintic Equation can be reduced to the form

$$x^5+px+q=0.$$

In 1834, G. B. Jerrard showed that a Tschirnhaus transformation can be used to eliminate the $x^{n-1}$, $x^{n-2}$, and $x^{n-3}$ terms for a general Polynomial equation of degree $n>3$.

See also Bring Quintic Form, Cubic Equation

References

Boyer, C. B. A History of Mathematics. New York: Wiley, pp. 472-473, 1968.

Tschirnhaus. Acta Eruditorum. 1683.