Pattern comments on Joy in Discovery: Galois theory

[Pattern]

There exists a polynomial p(x) of degree five over the rational numbers, with a root q, so that q cannot be obtained from the rationals by field operations and iterated extraction of roots.

This sounds like it could be done by finding one polynomial (of degree 5) which has this property. Is this accurate, or is the problem about solving $c_5\times x^5+c_4\times x^4+c_3\times x^3+c_2\times x^2+c_1\times x+c_0=0$ ?

[Eigil Rischel]

I added some clarification, but you are right.

(Since $x^5-10$ has the root ${}^5\sqrt{10}$, it’s clearly not true that all fifth-degree polynomials have this property)