Relative Densities of Ramified Primes 1 in Q( √ pq)

The relative densities of rational primes q, such that a given rational prime p is a square of either a principal or a non-principal ideal in the quadratic number field Q( √ pq), essentially depend on the residue classes of both p and q modulo 4. The computation of these densities is complete when at least one residue of p or q is not equal to 1 modulo 4. When both residues are equal to 1 modulo 4, it is shown that the densities are related to a problem of Legendre’s, and are evaluated taking as true an old conjecture on the solvability of the Pell equation x 2 − Ny 2 = −1. The proofs use elementary properties from algebraic number theory, and the conclusions on densities are a direct consequence of Dirichlet’s density theorem.