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.

in Lrather than in Q? $\endgroup$ – Pete L. Clark Jan 6 '10 at 18:584more comments