Entry 146

This is the quintic overview of Entries 140-144. Using the McKay-Thompson series \(j_n = j_n(\tau)\) for the Monster defined in Entry 145, if \(\tau\) are complex quadratics such that \(j_n(\tau)\) is a radical, then the following simple quintics have a solvable Galois group hence solvable in radicals $$\begin{align}x^5 + 5x^4 + 40x^3 &= j_1\\ x(x - 5)^4 &= j_2\\ x^3(x - 5)^2 &= j_3\\ x^5+5x &= \sqrt{\frac{64}{\sqrt{j_4}}-\sqrt{j_4}}\\ x^5 + 5x^3 - 10x^2 &= j_6\end{align}$$ Example. Let \(\tau = \tfrac{1+\sqrt{-163}}2\) so \(j_1(\tau) = -640320^3\). Let \(\tau = \tfrac{\sqrt{-58}}2\) so \(j_2(\tau) = 396^4\). Then $$x^5 + 5x^4 + 40x^3 = -640320^3\\ x(x - 5)^4 = 396^4$$ are quintics both solvable in radicals. Of course, it is well-known that $$\qquad e^{\pi\sqrt{163}} = 640320^3+743.99999999999925\dots\\ e^{\pi\sqrt{58}} = 396^4-104.00000017\dots$$There are infinitely many \(\tau\) but special ones such that \(j_n(\tau)\) are integers can be found in Entry 145.