Classifying the roots of polynomials with integer coefficients

Question

There is no generalized version of the quadratic formula to find the zeroes of polynomials with integer coefficients of degree $n>4$. I am curious about the forms that these zeroes take. More specifically, can they all be written as nested roots of rationals and rational complex numbers? For example something like $\sqrt{a-\sqrt[3]{b}}$? Or are more exotic constants like $e$ and $\pi$ sometimes appearing?

Answer 1

There is a theorem saying that it is impossible to solve any given polynomial of degree $n>4$ with just addition, subtraction, multiplication, division, and radicals. For example, the quintic $x^5-x+1=0$ cannot be expressed this way. However, you could use the Bring Radical, sometimes denoted as $\text{BR}(a)$, the solution to the polynomial $$x^5+x+a=0.$$ This is like how $\sqrt{a}$ solves $x^2-a=0$.

You could solve any quintic by reducing it to the form $$x^5+px+q.$$ using Tschirnhaus transformations. From here, you could solve this using $$\sqrt[4]{-\dfrac{p}{5}}\text{BR}\left(-\dfrac{1}{4}\left(-\dfrac{5}{p}\right)^{\frac{5}{4}}q\right).$$ Therefore, the possible roots for a quintic would be algebraic, meaning that $\pi$ and $e$ could not be solutions. This would also apply to higher degree polynomials.