An elementary problem in Galois theory about the roots of irreducible polynomials (Q6617375)

Let \(K\) be a number field and \(f\) an irreducible polynomial in \(K[x]\) of degree \(d\). Fix a root \(\alpha\) of \(f\). The number of roots of \(f\) that lie in in \(K(\alpha)\) is independent of \(\alpha\); denote this number by \(r_K(f)\). The main result of this article states:\N\NTheorem. If there exists a Galois extension \(L\) of degree \(n\) over \(K\) and \(L \cap K_f = K\) where \(K_f\) is the splitting field of \(f\). Then there exists an irreducible polynomial \(g \in K[x]\) of degree \(n \cdot d\) such that \(r_K(g) = n \cdot r_K(f)\).