The nth root function - $\sqrt[n]{a_{0}}$ - may be seen as an arithmetic operation (the inverse of the $pow(x, n)$ function) but it can also be interpreted as computing the roots of a specific class of polynomial - polynomials of the form $x^n - a_{0} = 0$ (which are cyclotomic if n is prime). Galois theory is a direct result of the fact that we only allow such roots (or "radicals") to have a closed form notation, whereas general polynomial roots are given no such notation and are thus, in general, "unsolvable by radicals" if the polynomial is of quintic degree or higher. Why is this?
As far as I am aware, such nth roots are not any easier or faster to calculate numerically than the roots of a general polynomial. The same methods are applied to both problems which involve the computation of irrational numbers. So what is special about nth roots as compared with the roots of other polynomials? Is there a logical reason that only such "pure" nth-root polynomial roots have a closed form or is just an accident of history?
In particular, I am considering a simple generalization of the nth root function from a binary function to a k+1-ary function that takes the last k coefficients of a polynomial as parameters, not just the very last coefficient. It would be an interesting question to see the resulting Galois theory for different values of k other than k = 1.
Note: The Bring radical or ultraradical is similar to this idea and can be used to solve otherwise unsolvable quintic equations, but is specific to quintic polynomials.