A Tschirnhausen Transformation can be used to algebraically transform a general Quintic Equation to the form
![\begin{displaymath}
z^5 + c_1 z + c_0 = 0.
\end{displaymath}](b_2257.gif) |
(1) |
In practice, the general quintic is first reduced to the Principal Quintic Form
![\begin{displaymath}
y^5 + b_2 y^2 + b_1 y + b_0 = 0
\end{displaymath}](b_2258.gif) |
(2) |
before the transformation is done. Then, we require that the sum of the third Powers of the
Roots vanishes, so
. We assume that the Roots
of the Bring-Jerrard
quintic are related to the Roots
of the Principal Quintic Form by
![\begin{displaymath}
z_i = \alpha {y_i}^4 + \beta {y_i}^3 + \gamma {y_i}^2 + \delta y_i + \epsilon.
\end{displaymath}](b_2262.gif) |
(3) |
In a similar manner to the Principal Quintic Form transformation, we can express the
Coefficients
in terms of the
.
See also Bring Quintic Form, Principal Quintic Form, Quintic Equation
© 1996-9 Eric W. Weisstein
1999-05-26