We can describe the roots of quadratic equations in terms of addition, subtraction, multiplication, division, and the square-root function $\sqrt a$ which computes a root of the special polynomial $x^2-a=0$. Similarly, the roots of cubic and quartic equations can be described with the aforementioned operations and the cube root $\root3\of a$.
In 1796, Bring found a method to express the roots of general quintic polynomials in terms of the aforementioned operations, the quintic root $\root5\of a$ and the bring radical $\mathop{\mathrm{Br}}(a)$ which computes a root of the special quintic polynomial $x^5+x+a$.
Can this scheme be extended? More specifically, can we express the roots of $n$-th degree polynomials in terms of roots of $r_q(a)$ of “special” polynomials of the form
$$r_q(a)=x_0\quad\hbox{such that }a+q_0+\sum_{k=1}^nq_kx^k=0?$$