On the Number of Times a Root off(n, x)=0 Generates a Field Containing a Given Number Field Original Research Article

Letf(x, y) be a polynomial with rational coefficients, and letEbe a number field. We prove estimates for the number of positive integersnless-than-or-equals, slantTsuch that some rootαoff(n, y)=0 satisfiesEsubset ofQ(α). The estimates are uniform with respect toEand, providedEsatisfies natural necessary conditions, take the formclog T/log log TT1/2, wherecdepends only onf. Under fairly general assumptions onf, we show that the factorT1/2may be removed. We also observe how these results lead to a version of Hilbertʹs Irreducibility Theorem uniform over number fields.