An Elementary Problem in Galois Theory about the Roots of Irreducible Polynomials

For a field $K$, and a root $\alpha$ of an irreducible polynomial over $K$ (in some algebraic closure) the number of roots of $f(x)$ lying in $K(\alpha)$ is studied here. Given such an $f(x)$ of degree $n$ for which $r$ of the roots are i n $K(\alpha)$, we describe a construction that yields, for $d\ge2$, irreducible polynomials 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 all number fields and possibly some more perfect fields. As an application, for $K=Q$ and positive integers $n\ge3,d\ge2$, we provide irreducible polynomials of degree $nd$ with exactly $d$ roots in the field generated by one of the roots. Independently, for $k<n$, we construct irreducible polynomials over the rationals of degree $n!/(n-k)!$ for which the field generated by one root contains exactly $k!$ roots. Many interesting new questions for further research are provided.