# Can we express the roots of all polynomials in terms of roots of some special polynomials?

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

$$r_q(a)=x_0\quad\hbox{such that }a+q_0+\sum_{k=1}^nq_kx^k=0?$$

• If you don't already know, it should be helpfully noted that the solutions to quintic equations are not always expressible in terms of radicals and arithmetic operations. – theREALyumdub Dec 4 '15 at 21:03
• @theREALyumdub It isn't with just ordinary radicals $\root n\of m$, but with the Bring-radical, you always can (as far as I understood). – FUZxxl Dec 4 '15 at 21:04
• Look up Galois theory. It was created for addressing questions like this one. I don't personally know the answer, but the question is essentially whether the combined splitting field of all polynomials of degree $n$ is a finite extension field of the rational numbers. – Paul Sinclair Dec 4 '15 at 23:36