### Solvable Quintics, Part 2

In the 1850’s, Jerrard showed that a Tschirnhausen transformation could reduce in radicals the general quintic into one missing three terms,

$\text{(1)}\;\; x^5+5ax+4b = 0$

In 1864, Bring independently would do the same. It is now known as the Bring-Jerrard quintic.  Such a reduction is important because it proved that a formula for the general quintic does exist, albeit it went beyond radicals and used elliptic functions, as was first done by Hermite.

In 1885, Runge et al showed that all solvable quintics [1] with rational coefficients have the form,

$x^5+5\,\frac{4u+3}{u^2+1}xz^4+4\,\frac{(2u+1)(4u+3)}{u^2+1}z^5 = 0$

A century later, Spearman and Williams gave their version,

$x^5+5\,\frac{4v+3}{v^2+1}xz^4+4\,\frac{-2v+11}{\,v^2+1}z^5 = 0$

So which is it? It turns out they are two sides of the same coin. Using the Spearman-Williams parametrization, let,

$a = \frac{4v+3}{v^2+1}z^4$

$b = \frac{-2v+11}{\,v^2+1}z^5$

and eliminating v between them using resultants, one gets,

$\text{(2)}\;\; 4az^6+8bz^5-5a^2z^2+2abz-b^2 = 0$

This is one of the simplest sextic resolvents for the quintic:  given {a, b}, if one can solve for z, then [1] is solvable. Since [2] is only a quadratic for b, we can easily solve for it,

$b = az+4z^5+2z\sqrt{(a+z^4)(-a+4z^4)}$

Still let,

$a= \frac{4v+3}{v^2+1}z^4$

and substituting it into the positive case of the square root yields the b of the 1885 version, while the negative case gives the b of the 1994 one, proving that they are indeed two sides of the same coin.

### A Tale of Three Solvable Octics

The following three cute irreducible octics are solvable,

$\text{(1)}\;\; x^8-5x-5 = 0$

$\text{(2)}\;\; x^8-44x-33 = 0$

$\text{(3)}\;\; x^8-x^7+29x^2+29 = 0$

However, each needs to be solved in a different way: they need a quadratic, quartic, and septic subfield, respectively.

The first is the easiest, it factors over $\sqrt{5}$ into two quartics.  The second does not factor over a square root extension, but factors into four quadratics,

$x^2+vx -(2v^3-7v^2+5v+33)/13 = 0$

where the coefficients are determined by the quartic,

$v^4+22v+22 = 0$

The command,

Resultant[x^2+vx -(2v^3-7v^2+5v+33)/13 , v^4+22v+22 ,v]

done in Mathematica or in www.wolframalpha.com will eliminate the variable v and recover [2].  The first two octics are by this author.  (Any other example for the second kind with small coefficients?)

The third (by Igor Schein) is the hardest, as it needs a septic subfield. Interestingly though, the solution involves the 29th root of unity. Given,

$x^8-x^7+29x^2+29 = 0$

Then,

\begin{aligned} x_1 &= (1+(a-b-c-d+e-f-g))/8\\ x_2 &= (1-(a-b-c-d-e+f+g))/8\\ x_3&= (1-(a+b-c+d+e-f+g))/8\\ x_4&= (1+(a+b-c+d-e+f-g))/8\\ x_5&= (1-(a+b+c-d+e+f-g))/8\\ x_6&= (1+(a+b+c-d-e-f+g))/8\\ x_7&= (1-(a-b+c+d-e-f-g))/8\\ x_8&= (1+(a-b+c+d+e+f+g))/8 \end{aligned}

where the 7 constants {a, b, c, d, e, f, g} is the square root of the appropriate root $z_i$ of the solvable septic,

$z^7-7z^6-2763z^5-19523z^4+1946979z^3+34928043z^2+\\119557031z-3247^2 = 0$

namely,

\begin{aligned} a &\approx \sqrt{ -26.98}\\ b &\approx \sqrt{ -26.95}\\ c &\approx \sqrt{ -19.71}\\ d &\approx \sqrt{ -4.78}\\ e &\approx \sqrt{ 0.08}\\ f &\approx \sqrt{ 36.91}\\ g &\approx \sqrt{ 48.43}\\ \end{aligned}

Note that,

$(8x_3-1)+(8x_4-1)+(8x_5-1)+(8x_6-1) = -4e$

hence one can use the roots of this octic to express the roots of its resolvent septic, and vice versa.  In terms of radicals, Peter Montgomery expressed the septic roots $z_i$ as,

\begin{aligned}\tfrac{1}{4}(z-1) &= 2(w^{11}+w^{13}+w^{16}+w^{18})-2(w+w^{12}+w^{17}+w^{28})\\ &+(w^3+w^7+w^{22}+w^{26})-(w^2+w^5+w^{24}+w^{27})\\ &+(w^4+w^{10}+w^{19}+w^{25})-(w^8+w^9+w^{20}+w^{21}) \end{aligned}

where w is any complex root of unity (excluding w = 1) such that $w^{29}=1$. For example, $w = \exp(2\pi i\cdot4/29)$ will yield the value for $z_1 \approx -26.98$,  and so on.

### A Family of Solvable Quintics and Septics

Define,

$x = \frac{-\sqrt{2}\,\eta(2\tau)}{\zeta_{48}\,\eta(\tau)}$

where $\eta$ is the Dedekind eta function, and $\zeta_{48}$ is the 48th root of unity.  Then for $\tau = \frac{1+\sqrt{-d}}{2}$ for d = {47, 103}, x is a root of the quintics,

$x^5-2x^4+2x^3-x^2+1 = 0$

$x^5-2x^4+3x^3-3x^2+x+1 = 0$

respectively. Note that the class number h(d) of both is 5.  It turns out these belong to a family of solvable quintics found by Kondo and Brumer,

$x^5-2x^4+2x^3-x^2+1 = nx(x-1)^2$

for any n, and where the two examples are n = {0, -1}.  A similar one for septics can be deduced from the examples in Kluner’s A Database For Number Fields as,

$x^7-2x^6+x^5-x^4-5x^2-6x-4 = n(x-1)x^2(x+1)^2$

with discriminant,

$d = 4^4(4n^3+99n^2+34n+467)^3$ .

The case n = 0 implies d = 467 and, perhaps not surprisingly, the class number of h(-467) = 7. However, since 467 does not have form 8m+7, then the eta quotient will be not be an algebraic number of degree h(-d).

To find a solvable family, it’s almost as if all you need is to find one right solvable equation, affix the right n-multiple of a polynomial on the RHS, and the whole family will remain solvable.

### Solvable quintics

Here is a nifty sufficient but not necessary condition on whether a quintic is solvable in radicals or not.  Given,

[1]  $x^5+10cx^3+10dx^2+5ex+f = 0$

If there is an ordering of its roots such that,

[2]  $x_1 x_2 + x_2 x_3 + x_3 x_4 + x_4 x_5 + x_5 x_1 - \\(x_1 x_3 + x_3 x_5 + x_5 x_2 + x_2 x_4 + x_4 x_1) = 0$

or alternatively, its coefficients are related by,

[3]  $-25c^6-40c^3d^2-16d^4+35c^4e+\\28cd^2 e -11c^2e^2+e^3-2c^2df-2def+cf^2 = 0$

then [1] is solvable as,

$x = z_1^{1/5}+z_2^{1/5}+z_3^{1/5}+z_4^{1/5}$

where the $z_i$ are the roots of the simple quartic,

$z^4+fz^3+(2c^5-5c^3e-4d^2e+ce^2+2cdf)z^2-c^5fz+c^{10} = 0$

Note that [3] in fact is the constant term of Cayley’s resolvent sextic and is only quadratic in f.  Using another relation among the $x_i$, Dummit’s resolvent has a constant term that is already a quartic in f, hence the choice of relation matters.

Example 1: This family of quintics by this author satisfies [3],

$x^5+10x^3+5(n^2+3n+18)x^2-5(n^3+n^2+15n-14)x+\\(n^4-n^3+37n^2+441) = 0$

Let n = 1, and we have,

$x^5+10x^3+110x^2-15x+478 = 0$

Quartic is,

$z^4+478z^3+11994z^2-478z+1 = 0$

such that,

$x = z_1^{1/5}+z_2^{1/5}+z_3^{1/5}+z_4^{1/5} = -4.50991\dots$

Example 2: Another good example of [3] is Emma Lehmer’s quintic,

$y^5 +n^2y^4-(2n^3+6n^2+10n+10)y^3+\\(n^4+5n^3+11n^2+15n+5)y^2+(n^3+4n^2+10n+10)y+1$

The linear transformation,

$y = x-n^2/5$

will reduce it into the form of [1], and it will then be seen its coefficients obey [3]. As a particular example, let n = 5 and we have the reduced form,

$x^5-710x^3+11005x^2-59640x+108701 = 0$

Let its roots be,

$(x_1, x_2, x_3, x_4, x_5) \approx (-33.15,\; 4.83,\; 12.16,\; 4.99,\; 11.16)$

and we find that indeed it obeys [2].

Unfortunately, no similar simple relation between the coefficients of a solvable septic, or 7th degree equation, is yet known.