Entry 196

In Entry 190 and Entry 191, two relations between solvable quintics and \(G_m\) and \(g_m\) were proposed. Going higher,

Conjecture: The following sextics have a solvable Galois group

$$\begin{align}x^6+10x^3+5 &= 4\left(\frac{4}{G_{m}^{16}}-G_{m}^{8}\right)x \\ x^6+10x^3+5 &= 4\left(\frac{4}{g_{m}^{16}}+g_{m}^{8}\right)x \end{align}$$

For example, \(G_5 = \left(\tfrac{1+\sqrt5}2\right)^{1/4}\) and \(g_{10} = \left(\tfrac{1+\sqrt{5}}2\right)^{1/2}\) yields

$$\begin{align}x^6+10x^3+5 &= -2\left(-25+13\sqrt5\right) x \\ x^6+10x^3+5 &= -6\left(-65+27\sqrt5\right) x \end{align}$$

which are solvable in radicals, and so on.