The distribution of the multiplicative index of algebraic numbers over residue classes (Q6564794)

Let \(K\) be a number field and let \(F/K\) be a finite Galois extension. Let \(G\) be a finitely generated torsion free subgroup of \(K^{\times}\). For a prime \(\mathfrak p\) of \(K\), we denote by \(\mathrm{ind}_{\mathfrak p}(G)\) the index of the image of \(G\) in the multiplicative group of the residue field at \(\mathfrak p\). Let \(S\) be a non empty set of positive integers. Let \(C\) be a union of conjugacy classes of the Galois group of \(F/K\). Under the Generalized Riemann Hypothesis, the authors find the distribution of the primes \(\mathfrak p\) of \(K\) for which \N\[\N\mathrm{ind}_{\mathfrak p}(G)\in S\text{ and }\mathrm{Frob}_{F/K}(\mathfrak p)\in C. \N\]