Composing radicals can only express certain types of permutation groups, and it's not too hard to see how it works. The quadratic formula term `-b +- sqrt(a^2 - 4ac)` is basically exploiting the permutation created by `sqrt(b)` which swaps between `+sqrt(b)` and `-sqrt(b)`. A cube root `cbrt(b)` rotates between three values, which are separated by roots of unity `e^(2pi/3 i)`. A fourth root swaps between four values, which are separated by the fourth roots of unity `(1, i, -1, -i)`. Etc.
So the basic "building block" is "rotate 2 points", "rotate 3 points", "rotate 4 points", etc. Even those doesn't express very much: a generic cubic has symmetry S_3, which is arbitrary permutations of three roots, and even that can't be expressed with just basic sums of these, because the only way to get three-fold symmetry is to use a single cube root.
But take a look at the cubic formula: https://en.wikipedia.org/wiki/Cubic_equation#General_cubic_f.... Turns out if you nest them you can create more interesting symmetries. The cubic formula structure has the structure "sums of cube roots of square roots", which give six total values but really it makes three because of cancellation.
I interpret this to mean that as you try to make the larger permutation groups you're increasingly stressing the expressive powers of radicals. The quadratic did fine with two: the quadratic formula is basically "take the midpoint of the roots -b/2a and add the offset between the two `sqrt((b/2a)^2 - c)` with a plus or minus sign". The cubic formula gets messy because the roots don't have to be evenly spaced so you need a way to express that too. The quartic formula is insane (https://www.curtisbright.com/quartic/quartic-png.html) and involves sums of square roots of sums of third roots of square roots. So it is not too surprising that things fall apart soon after, apparently at the quintic.
The specific way that the quintic fails is that the symmetry group S_5 is not "solvable", which is hard to explain without a lot more space, but which basically means that it includes a list of operations (P, Q, ...) which when you replace them with their commutators (PQP^-1 Q^-1, ...) don't get any simpler. I guess that is what radicals don't do: commuting radicals always makes them simpler, and so at N=5 you hit a group they can't fake.
Composing radicals can only express certain types of permutation groups, and it's not too hard to see how it works. The quadratic formula term `-b +- sqrt(a^2 - 4ac)` is basically exploiting the permutation created by `sqrt(b)` which swaps between `+sqrt(b)` and `-sqrt(b)`. A cube root `cbrt(b)` rotates between three values, which are separated by roots of unity `e^(2pi/3 i)`. A fourth root swaps between four values, which are separated by the fourth roots of unity `(1, i, -1, -i)`. Etc.
So the basic "building block" is "rotate 2 points", "rotate 3 points", "rotate 4 points", etc. Even those doesn't express very much: a generic cubic has symmetry S_3, which is arbitrary permutations of three roots, and even that can't be expressed with just basic sums of these, because the only way to get three-fold symmetry is to use a single cube root.
But take a look at the cubic formula: https://en.wikipedia.org/wiki/Cubic_equation#General_cubic_f.... Turns out if you nest them you can create more interesting symmetries. The cubic formula structure has the structure "sums of cube roots of square roots", which give six total values but really it makes three because of cancellation.
I interpret this to mean that as you try to make the larger permutation groups you're increasingly stressing the expressive powers of radicals. The quadratic did fine with two: the quadratic formula is basically "take the midpoint of the roots -b/2a and add the offset between the two `sqrt((b/2a)^2 - c)` with a plus or minus sign". The cubic formula gets messy because the roots don't have to be evenly spaced so you need a way to express that too. The quartic formula is insane (https://www.curtisbright.com/quartic/quartic-png.html) and involves sums of square roots of sums of third roots of square roots. So it is not too surprising that things fall apart soon after, apparently at the quintic.
The specific way that the quintic fails is that the symmetry group S_5 is not "solvable", which is hard to explain without a lot more space, but which basically means that it includes a list of operations (P, Q, ...) which when you replace them with their commutators (PQP^-1 Q^-1, ...) don't get any simpler. I guess that is what radicals don't do: commuting radicals always makes them simpler, and so at N=5 you hit a group they can't fake.