In general, polynomials of degree 5 or higher, with rational coefficients, are not solvable by radicals. There are some exceptions and trivial cases, for example $x^5-5x+12$ and $x^{100}+x^{25}+3$, respectively.
All irreducible polynomials with rational coefficients, that I am aware of, have either none or all roots that can be expressed by radicals.
Do partially-solvable irreducible polynomials exist? If they do, are there any examples known?
I mean, for example, an irreducible polynomial of $n$th degree that can be split over $Q(\sqrt{2})$ into two polynomials of 3rd and $(n-3)$th degrees, where the polynomial of $(n-3)$th is not solvable by radicals.
At https://math.stackexchange.com/q/662972 Alexander Gruber wrote:
Polynomials which are not solvable by radicals have (at least one) root that cannot be written by any combination of the operations of addition, multiplication, and the taking of nth roots.
I am not sure if he meant only reducible polynomials, and/or with non-rational coefficients, as polynomials that have "at least one" root that cannot be expressed by radicals.