Editing Quintic function (section)

===Quintics in Bring–Jerrard form===

There are several parametric representations of solvable quintics of the form {{math|''x''<sup>5</sup> + ''ax'' + ''b'' {{=}} 0}}, called the [[Bring–Jerrard form]].

During the second half of the 19th century, John Stuart Glashan, George Paxton Young, and [[Carl Runge]] gave such a parameterization: an [[irreducible polynomial|irreducible]] quintic with rational coefficients in Bring–Jerrard form
is solvable if and only if either {{math|''a'' {{=}} 0}} or it may be written
:<math>x^5 + \frac{5\mu^4(4\nu + 3)}{\nu^2 + 1}x + \frac{4\mu^5(2\nu + 1)(4\nu + 3)}{\nu^2 + 1} = 0</math>
where {{math|''μ''}} and {{math|''ν''}} are rational.

In 1994, Blair Spearman and Kenneth S. Williams gave an alternative,
:<math>x^5 + \frac{5e^4( 4c + 3)}{c^2 + 1}x + \frac{-4e^5(2c-11)}{c^2 + 1} = 0.</math>

The relationship between the 1885 and 1994 parameterizations can be seen by defining the expression
:<math>b = \frac{4}{5} \left(a+20 \pm 2\sqrt{(20-a)(5+a)}\right)</math>
where {{tmath|1=a=5\tfrac{4\nu+3}{\nu^2+1} }}. Using the negative case of the square root yields, after scaling variables, the first parametrization while the positive case gives the second.

The substitution {{tmath|1=c = - \tfrac{m}{\ell^5}, }} {{tmath|1=e = \tfrac{1}{\ell} }} in the Spearman–Williams parameterization allows one to not exclude the special case {{math|''a'' {{=}} 0}}, giving the following result:

If {{mvar|a}} and {{mvar|b}} are rational numbers, the equation {{math|''x''<sup>5</sup> + ''ax'' + ''b'' {{=}} 0}} is solvable by radicals if either its left-hand side is a product of polynomials of degree less than 5 with rational coefficients or there exist two rational numbers ''ℓ'' and {{mvar|m}} such that
:<math>a=\frac{5 \ell (3 \ell^5-4 m)}{m^2+\ell^{10}}\qquad b=\frac{4(11 \ell^5+2 m)}{m^2+\ell^{10}}.</math>