On the number of times a root of \(f(n,x)=0\) generates a field containing a given number field
Let \(f(x,y)\) be a polynomial with rational coefficients. The author obtains upper bounds for the number \(N(T)\) of integers \(n\leq T\) such that there exists a root of \(f(n,y)=0\) generating a field which contains a given algebraic number field \(E\). In the case when \(f\) is irreducible over the rationals and monic in \(y\) and \(E\) is not contained in the algebraic closure \({\mathcal C}\) of the rationals in the splitting field \(\Lambda\) of \(f\) over \(\mathbb{Q}(x)\), then \(N(T)\) does not exceed \(c^{\log T/\log\log T}\sqrt T\) with \(c\) being dependent only on \(f\) but not on \(E\). If moreover every intermediate field \({\mathcal C}(x)\subset\Omega\subset\Lambda\) (\(\Omega\neq{\mathcal C}(x)\)) has positive genus and is ramified above at least one irrational point then the factor \(\sqrt T\) in this bound can be omitted. As an application one obtains a uniform version of Hilbert's irreducibility theorem for number fields.
scientific article; zbMATH DE number 1229751
scientific article