Entry 197

From Entry 195

$$2^{-1/4}G_{71} = x,\quad \text{where}\, x^7 - 2x^6 - x^5 + x^4 + x^3 + x^2 - x - 1=0$$

We will now give the radical solution to this septic using the well-known cubic \(r^3+r^2-2r-1=0\) with roots $$r_1,\,r_2,\,r_3 = 2\cos\big(\tfrac{2\pi}7\big),\, 2\cos\big(\tfrac{4\pi}7\big),\, 2\cos\big(\tfrac{6\pi}7\big)$$Define the function

$$\begin{align}y_n &= P(r_n) =\frac{29323 r^2 - 20538 r - 15494 + (1193 r^2 - 1048 r - 730)\sqrt{7\times71}}2\\ y_{n+3} &= Q(r_n) =\frac{29323 r^2 - 20538 r - 15494 - (1193 r^2 - 1048 r - 730)\sqrt{7\times71}}2\end{align}$$

For example, \(y_1 = P(r_1) \approx 219.6454\). Then the real root of the septic is

$$\quad x = \frac{2 + y_1^{1/7}+y_2^{1/7}+y_3^{1/7}+y_4^{1/7}+y_5^{1/7}+y_6^{1/7}}7 = 2.1306068\dots$$

In fact, the \(y_n\) are the roots of a sextic with rather large coefficients,

$$y^6+ay^5+by^4+cy^3+dy^2+ey-461^7 = 0$$

and where the constant term is a \(7\)th power. This sextic is special since it can factor either over the square root extension \(\sqrt{7\times71}\) or the cubic extension \(2\cos\big(\tfrac{2\pi}7\big)\).