An elementary problem in Galois theory about the roots of irreducible polynomials

For a field K, and a root \(\alpha \) $\alpha$ of an irreducible polynomial over K (in some algebraic closure \(\bar{K}\) $\overline{K}$ ) the number of roots of f(x) lying in \(K(\alpha )\) $K(\alpha )$ is studied here. Given such an f(x) of degree n for which r of the roots are in \(K(\alpha )\) $K(\alpha )$ , we describe a construction that yields, for \(d\ge 2\) $d\ge 2$ , irreducible polynomial g(x) of degree nd and with exactly rd of the roots in the field generated by any one root of those polynomials. Our results are valid for fields of characteristic 0 and possibly all infinite perfect fields. As an application, for any algebraic number field K and positive integers \(n\ge 3,d\ge 2\) $n\ge 3,d\ge 2$ , we provide irreducible polynomials of degree nd with exactly d roots in the field generated by one of the roots. Independently, for \(k<n\) $k<n$ , we prove directly the existence of irreducible polynomials over \(\textbf{Q}\) $\mathbf{Q}$ of degree \(n!/(n-k)!\) $n!/(n-k)!$ for which the field generated by one root contains exactly k! roots. Many interesting new questions for further research are provided.