Ugly solutions to easily stated problems

Question

I recently saw a very hideous closed form for a quartic equation here: Does a closed form solution exist for $x$?

For fun, I'm wondering about surprisingly ugly solutions/ complicated machinery needed to problems that are simply stated.

Clearly, they all don't have to be algebra-based.

I'm trying to get a sense for how particular different mathematic methods are necessary, and interesting. Even for easily stated problems.

Answer 1

Classify pairs of commuting matrices up to a change of basis.

This sounds like an extension of Jordan normal form, since the matrices should be simultaneously Jordan-izable or something.

In fact it is the canonical example of a "wild" linear algebra problem that is as hard as classifying $k$-tuples of (noncommuting) matrices for all $k$. No reasonable solution is expected to exist.

Answer 2

Given a polynomial, $p(x)$, with integer coefficients, what numbers, $z$, satisfy the equation $p(z)=0$.

In the case of small degree polynomials, we have the ability to write down the solutions to polynomial equations with integer coefficients using integers, the symbols $(,),+,-,\cdot,/,$ and ^, where the last indicates exponentiation. In degree $2$, this gives rise to the well known equation $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

In degree 5 or greater, this is suddenly no longer true. There are fifth-degree polynomials with solutions that cannot be exactly described in any "reasonable" way.