I know that the Galois group of an equation on $mathbb{Q}$ can be determined according to the rational relation determined by Vieta’s theorem.

Page 393 of this book also points out that these relations are to keep all polynomial relations between root sets (the result are all rational numbers) unchanged:

Taking the equation $f(x)=x^{4}-4x^{2}-5$ as an example, its four roots are $sqrt{5}$, $-sqrt{5}$, $i$ and $-i$ in turn.

```
f1(x1_, x2_, x3_, x4_) := x1 + x2
f2(x1_, x2_, x3_, x4_) := x3 + x4
f3(x1_, x2_, x3_, x4_) := x1*x2(*Rational relations between root sets determined by Vieta theorem*)
G = SymmetricGroup(4);
sol = Select({Permute(Inactive(f1)(x1, x2, x3, x4), G),
Permute(Inactive(f2)(x1, x2, x3, x4), G),
Permute(Inactive(f3)(x1, x2, x3, x4),
G)}(Transpose), ((((Activate(#) /. {x1 ->
Root(#1^4 - 4 #1^2 - 5 &, 1),
x2 -> Root(#1^4 - 4 #1^2 - 5 &, 2),
x3 -> Root(#1^4 - 4 #1^2 - 5 &, 3),
x4 -> Root(#1^4 - 4 #1^2 - 5 &, 4)})) //
FullSimplify) == {0, 0, -5}) &)
FindPermutation(#, Inactive(f1)(x1, x2, x3, x4)) & /@ sol((All, 1))
PermutationList(#, 4) & /@ %(*The group is isomorphic to the Klein's four-group and is a subgroup of group S4*)
```

From the above code, we can see that by keeping the polynomial relationship between the root sets such as $x1 + x2 = 0$, $x3 +x4 = 0$, $x1*x2=-5$, which results with rational numbers unchanged, we find the Galois group of the equation $f(x)=x^{4}-4x^{2}-5$ on $mathbb{Q}$.

If the Galois group `G`

is solvable, then the equation $f(x)$ on $mathbb{Q} $ is radical solvable and the group has a normal subgroup sequence $mathrm{G}=mathrm{G}_{0} triangleright mathrm{G}_{1} triangleright ldots triangleright mathrm{G}_{mathrm{r}}={mathrm{e}}$. Let every quotient group $G_{i} / G_{i+1}, i=0,1,2, ldots, r-1$ be commutative.

I want to know what symmetric property of the root set of the representation of this Galois group `G`

determines that it can have radical solutions. And what property of the Galois group described by the symmetry of the root set determines that it can be solved by radical. In other words, which of the more in-depth properties between the root sets represented by the Galois group determine that the equation can be solved by radical. It’s better to show this relationship clearly with MMA.