A converse of Artin's density theorem: The case of cubic fields (Q687541)

For fixed \(n\) consider the sequence \({\mathcal A}\) of all algebraic integers of degree \(n\), ordered according to the height of the minimal polynomials. For any prime \(p\) consider the subsequence \({\mathcal A}(p,\Phi)\) of \({\mathcal A}\) formed by algebraic integers generating a field in which \(p\) factorizes into prime ideals in a prescribed manner \(\Phi\). The authors show that \({\mathcal A}(p,\Phi)\) has always a natural density in \({\mathcal A}\) and compute it explicitly in the case of \(n=3\).