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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 rootsare 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 inthe field generated by any one root of those polynomials. Our results are validfor all number fields and possibly some more perfect fields. As an application,for $K=Q$ and positive integers $n\ge3,d\ge2$, we provide irreduciblepolynomials of degree $nd$ with exactly $d$ roots in the field generated by oneof the roots. Independently, for $k<n$, we construct irreducible polynomialsover the rationals of degree $n!/(n-k)!$ for which the field generated by oneroot contains exactly $k!$ roots. Many interesting new questions for furtherresearch are provided.

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