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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?

Anon21
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    A polynomial is soluble by radicals iff its Galois group is soluble. Lots is known about computing Galois groups of quintics, e.g. https://mathoverflow.net/questions/80574/galois-groups-at-closed-points-from-galois-group-at-generic-point and https://math.stackexchange.com/questions/38896/calculating-the-galois-group-of-an-irreducible-quintic I think the generic (in a suitable sense) Galois group is $\mathfrak{S}_5$, i.e. is not soluble. – Pulcinella Jun 07 '20 at 21:34
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    Some trivial remarks: You should probably work over $\mathbf{Q}$ because over $\mathbf{C}$ all polynomials split so their Galois groups are trivial. All polynomials are realisable as the characteristic polynomials of some matrix, via companion matrices. A reasonable question might be what relations there are between the Galois group and other properties of the matrix, e.g. facts about its minors. – Pulcinella Jun 07 '20 at 21:35
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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.

  • Can you explain the need to limit the argument to the rationals. Everything I can find about the Abel–Ruffini theorem seems to suggest it applies to the reals. So I do not know why you need this limitation? – Anon21 Aug 09 '20 at 13:14