Connection between Abel–Ruffini theorem and characteristic polynomial of matrices Suppose $n \times n$ matrix $M$ with arbitrary coefficient in $\mathbb{R}$ or $\mathbb{C}$.
In the general case, the characteristic polynomial of $M$ is a polynomial whose highest degree is $n$. 
Is there a link between $n>4$ and the Abel–Ruffini theorem?
https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem
Are the roots of a general $5 \times 5$ matrix subject to the Abel–Ruffini theorem limitations?
What requirements on $M$ must there be for its roots to be subject to the Abel–Ruffini theorem? Is it sufficient that the entries of $M$ be arbitrary? 
 A: If a matrix $A\in M_n(\mathbb{Q})$ is random, then roughky speaking, its characteristic polynomial is random. Then we may consider a (non monic) random polynomial $p=\sum_{0\leq i\leq 5}a_ix^i\in \mathbb{Z}_5[x]$.
The simplest method is to consider a positive integer $n$ and to randomly choose (independently) the $(a_i)$ uniformly in $\{-n..n\}$. Let $P_n$ be the associated probability that $p$ is irreducible and has $S_5$ as Galois group. 
EDIT. Then $\lim_{n\rightarrow +\infty}P_n=1$. About this result, you can read 
[1] J.P. Serre: Topics in Galois Theory.(the reading is hard)
[2] Igor Irvine: Galois groups of generic polynomials.
https://arxiv.org/pdf/1511.06446.pdf
A difficult problem is to estimate the speed of convergence towards $1$ of $P_n$; an upper bound is given in [1] and more precisely in [2].
To give an idea, here are the results of some random tests
$P_1\approx 28$%,$P_{10}\approx 85$%$,P_{100}\approx 98.2$%,$P_{1000}\approx 99.79$%.
Of course, $P_n$ depends on the degree of the polynomial $p$. When the degree increases, $P_n$ increases too.
