A Density of Ramified Primes

Let K be a cyclic number field of odd degree over [Formula: see text] with odd narrow class number, such that 2 is inert in [Formula: see text] . We define a family of number fields [Formula: see text] , depending on K and indexed by the rational primes p that split completely in [Formula: see text] , 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 [Formula: see text] is strictly between 0 and 1 and is given explicitly as a formula in terms of the degree of the extension [Formula: see text] . Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals.