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.