The irreducible but solvable octic,

$$x^8-x^7+29x^2+29=0\tag{1}$$

was first mentioned by Igor Schein in this 1999 sci.math post. This does not factor over a quadratic or quartic extension, but over a 7th deg one. It can also be nicely solved using the $29th$ root of unity. Let $\omega = \exp(2\pi i /29)$ then define,

$$y = y_k = \omega^{k}+\omega^{12k}+\omega^{17k}+\omega^{28k}\tag{2}$$

$$z_k = 4(y^3+y^2-9y-4)(y^2-2)(y-1)+9\tag{3}$$

then I found a pair of roots of $(1)$ as,

$$x = \frac{1\color{red}{-}\sqrt{z_{1}}+\sqrt{z_{2}}+\sqrt{z_{4}}+\sqrt{z_{8}}+\sqrt{z_{16}}+\sqrt{z_{32}}+\sqrt{z_{64}}}{8} \approx 1.79106+0.8286\,i\dots$$

$$x = \frac{1+\sqrt{z_{1}}\color{red}{-}\sqrt{z_{2}}\color{red}{-}\sqrt{z_{4}}+\sqrt{z_{8}}+\sqrt{z_{16}}\color{red}{-}\sqrt{z_{32}}+\sqrt{z_{64}}}{8} \approx 1.79106-0.8286\,i\dots$$

and the other pairs using appropriate signs of the square roots.

*Note*: Of course, $y_k$ and $z_k$ are roots of two different 7th-deg eqns with integer coefficients, while $(3)$ is the 6th-deg Tschirnhausen transformation between them. (In an earlier edit, I used an alternative expression for $z_k$ by P. Montgomery found in the sci.math link, but I like this one better.)

* Question*: Does anyone know why $(1)$ has such a

*simple*form, and if we can find other similar irreducible but solvable octics involving a $p$th root of unity for other prime $p$? (For some reason, this does not appear in the Kluener's database of number fields for 8T25.)

maximato $$ \text{factor(X^8-X^7+29*X^2+29, x^8-4*x^7+8*x^6-6*x^5+2*x^4+6*x^3-3*x^2+x+3);} $$ and observed the linear factor in the numerator. I don't know what algorithmmaximauses, but it might well be integer relations. – Noam D. Elkies Oct 18 '13 at 4:35