Let $L$ be a Galois number field over $\mathbb{Q}$. Based on classical algebraic number theory (specifically, the Chebotarev density theorem), I can answer lots of numerical questions about how primes $p\in\mathbb{Z}$ split in $\mathfrak{O}_L$. For example the probability that $p$ splits completely is $1/[L:\mathbb{Q}]$, and the average number of factors is the sum over the Galois group of the reciprocals of the orders (so, for a quadratic extension, we get $1+1/2=3/2$ since half the primes split, and half don't.
My question: Another question one can ask is if the factors of $p$ are principal (since $L$ is Galois, they are all conjugate, and thus all principal or all not). Are there any quantitative results about this?
One might optimistically hope that the probability of this is the reciprocal of the class number, but I have no idea if this is true.
Best Answer
Yes, the density is 1/h. Apply Chebotarev to the Hilbert class field of L.
Edit: As Pete points out, this is not right if I am counting primes in Q. It is only correct if I count prime ideals of L and, as Kevin points out, this only sees the primes in Q splitting completely in L.
Edit again: Isn't the Hilbert class field of L galois over Q with the Galois group being a semi-direct product of Gal(L/Q) and the class group? If that's the case, then we apply Chebotarev to the Hilbert class field of L as an extension of Q and the density is whatever comes out. So for L/Q quadratic, for example, the density is 1/2 + 1/2h. The 1/2 comes from the inert primes and the 1/2h from the split principal primes.