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Polynomials of degree larger than 4 with integer coefficients may have non-algebraic solutions, that is, zeros that do not have an algebraic expression (over the integers).

How likely is this to happen?

For example, of the $(2M+1)^{n}$ polynomials of degree less than or equal to $n\geq 5$ with coefficients in $\{-M,\dots,M\}$, what is the asymptotic behavior, as $M\to\infty$, of the proportion of those polynomials with no/at least one/only algebraic solutions?

Bananach
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  • The word "algebraic" is confusing in this context (although it seems it is used). All the solutions will be algebraic numbers (as opposed to transcendentals). They cannot be expressed with radicals. – quid Nov 19 '18 at 21:04
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    @quid That's what I thought until I saw the wikipedia article. I like the succinctness though, and that it will attract people to this question that think "Ha, I can answer that trivial question". Nevertheless, that's why I clarified in the first sentence of the question – Bananach Nov 20 '18 at 08:19

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