Bring-Jerrard Quintic Form

A Tschirnhausen Transformation can be used to algebraically transform a general Quintic Equation to the form

$$z^5 + c_1 z + c_0 = 0.$$ (1)

In practice, the general quintic is first reduced to the Principal Quintic Form

$$y^5 + b_2 y^2 + b_1 y + b_0 = 0$$ (2)

before the transformation is done. Then, we require that the sum of the third Powers of the Roots vanishes, so $s_3(y_j) = 0$. We assume that the Roots $z_i$ of the Bring-Jerrard quintic are related to the Roots $y_i$ of the Principal Quintic Form by

$$z_i = \alpha {y_i}^4 + \beta {y_i}^3 + \gamma {y_i}^2 + \delta y_i + \epsilon.$$ (3)

In a similar manner to the Principal Quintic Form transformation, we can express the Coefficients $c_j$ in terms of the $b_j$.

See also Bring Quintic Form, Principal Quintic Form, Quintic Equation