Entry 195

Continuing Entry 194 for \(G_d\) with \(d \equiv 7\,\text{mod}\,8\)

IV. n = 7

There are only \(d =71, 151, 223, 463, 487\) though, to prevent clutter, we include only the first two 

$$\begin{align}G_{71} &= 2^{1/4}x,\quad x^7 - 2x^6 - x^5 + x^4 + x^3 + x^2 - x - 1=0\\ G_{151} &= 2^{1/4}x,\quad x^7 - 3x^6 - x^5 - 3x^4 - x^2 - x - 1=0 \end{align}$$ It's remarkable how small the coefficients are. We will solve these in radicals in another entry.

V. n = 9

There are only \(d =199, 367, 823, 1087, 1423\) though again the first two 

$$\begin{align}G_{199} &= 2^{1/4}x,\quad x^3 - (r^2 + 3r + 1)x^2 - x + r = 0,\quad\qquad r^3 + 4r^2 + r + 1 = 0\\ G_{367} &= 2^{1/4}x,\quad x^3 - (r^2 + 5r + 1)x^2 - (2r + 1)x + r =0,\quad r^3 + 7r^2 + 4r + 1 = 0 \end{align}$$

As nonics, these also have small coefficients and are easier solved in radicals as they can be factored over a cubic extension.