Sₙ-extensions with prescribed norms

Given a number field k, a finitely generated subgroup A⊆ k^×, and an integer n≥ 3, we study the distribution of Sₙ-extensions of k such that the elements of A are norms. For n≤ 5, and conjecturally for n ≥ 6, we show that the density of such extensions is the product of so-called ``local masses'' at the places of k. When n is an odd prime, we give formulas for these local masses, allowing us to express the aforementioned density as an explicit Euler product. For $n=4$, we determine almost all of these masses exactly and give an efficient algorithm for computing the rest, again yielding an explicit Euler product.