How to solve this solvable 8th-degree algebraic equation by radicals? 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

 A: 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.
A: A Pari computation shows that the discriminant of the degree $8$ number field $F$ generated by a root of the polynomial $f=^8−8^7+8^6+40^5−14^4−232^3+488^2−568+1$ is the product of a power of $2$  and a power of $3$.  Therefore any quadratic subfield of $F$ is of the form ${\bf Q}(\sqrt{\pm d})$ with $d$ a divisor of $6$. A Pari computation shows that $f$ factors into a  product of two degree $4$ irreducible factors over ${\bf Q}(\sqrt{2})$ and over ${\bf Q}(\sqrt{3})$. This implies that we have ${\bf Q}(\sqrt{2},\sqrt{3})\subset F$. A final Pari computation shows that $f$ is a product of four quadratic polynomials in  ${\bf Q}(\sqrt{2},\sqrt{3})[X]$. Their roots are equal to $1+\sqrt{2}+{\root 4 \of 3}(1+{\root 4 \of 3})$ and its conjugates.
