TITLE: How to solve this solvable 8th-degree algebraic equation by radicals?
QUESTION [17 upvotes]: Solve the following equation in radicals.
$$x^8-8x^7+8x^6+40x^5-14x^4-232x^3+488x^2-568x+1=0$$
I use Magma to verify that its Galois group is a solvable group.
R := RationalField(); 
R < x > := PolynomialRing(R); 
f := x^8-8*x^7+8*x^6+40*x^5-14*x^4-232*x^3+488*x^2-568*x+1; 
G := GaloisGroup(f); 
print G;
GroupName(G: TeX:=true);
IsSolvable(G);

The output of Magma(Online) is:
Permutation group G acting on a set of cardinality 8
Order = 16 = 2^4
    (2, 4)(6, 8)
    (1, 2, 3, 4)(5, 6, 7, 8)
    (1, 5)(2, 8)(3, 7)(4, 6)
C_2\times D_4
true

I also tried to calculate with PARI/GP(64-bit)v_2.13.3+GAP(64-bit)v_4.11.1, but failed.
gap> LoadPackage("radiroot");
true
gap> x := Indeterminate(Rationals,"x");;
gap> g := UnivariatePolynomial( Rationals, [1,-8,8,40,-14,-232,488,-568,1]);
x^8-8x^7+8x^6+40x^5-14x^4-232x^3+488x^2-568x+1
gap>  RootsOfPolynomialAsRadicals(g, "latex");
"/tmp/tmp.sfoZ6C/Nst.tex"
Error，AL_EXECUTABLE，the executable for PARI/GP，has to be set at /proc/
cygdrive/C/gap-4.11.1/pkg/aInuth-3.1.2/gap/kantin.gi : 205 called from

REPLY [8 votes]: Just to explain why the attempt to calculate in GAP failed. The error message was produced by the Alnuth GAP package on which RadiRoot depends, and Alnuth in its turn requires PARI/GP. If everything is installed, the code in question worked (also note the correct order of the coefficients):
gap> LoadPackage("radiroot");
true
gap> x := Indeterminate(Rationals,"x");;
gap> g := UnivariatePolynomial( Rationals, [1,-568,488,-232,-14,40,8,-8,1]);
x^8-8*x^7+8*x^6+40*x^5-14*x^4-232*x^3+488*x^2-568*x+1
gap> RootsOfPolynomialAsRadicals(g, "latex");
"/var/folders/dt/some_random_path/tempfilename.tex"

It wrote a temporary file and displayed its name. The file contains all commands needed to compile it with LaTeX, and I just show here the crucial part of the output (without modifying the content, so it has some redundant details) which says the following:

An expression by radicals for the roots of the polynomial $$x^{8} - 8x^{7} + 8x^{6} + 40x^{5} - 14x^{4} - 232x^{3} + 488x^{2} - 568x + 1$$ with the $n$-th root of unity $\zeta_n$ and
$$\omega_1 = \sqrt[2]{2},$$
$$\omega_2 = \sqrt[2]{3},$$
$$\omega_3 = \sqrt[2]{-\omega_2},$$
$$\omega_4 = \sqrt[2]{\omega_2},$$
is:
$$1-\omega_1-\omega_2-\omega_3$$

For other sources of help with GAP, especially with specific GAP packages, please see the GAP tag description.