A density of ramified primes

Let K be a cyclic number field of odd degree over 𝑄 with odd narrow class number, such that 2 is inert in 𝐾/𝑄. We define a family of number fields {𝐾(𝑝)}𝑝, depending on K and indexed by the rational primes p that split completely in 𝐾/𝑄, in which p is always ramified of degree 2. Conditional on a standard conjecture on short character sums, the density of such rational primes p that exhibit one of two possible ramified factorizations in 𝐾(𝑝)/𝑄 is strictly between 0 and 1 and is given explicitly as a formula in terms of the degree of the extension 𝐾/𝑄. Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals.