If $K$ is a number field, a result from Bach tells us that the primes in $K$ of norm smaller that $12 (\log |\mathrm{Discriminant}(K)|)^2$ generate the ideal class group $\mathrm{Cl}\_K$. Is there any known bound if we require the generating primes to be unramified in $K/\mathbb Q$? (i.e. a result of the form "prime ideals in $K$ above the prime numbers unramified in $K/\mathbb Q$ and smaller than $f(K)$ generate the ideal class group")


In the particular case I study, $[K:\mathbb Q] = 4$. With algebraic methods I can find a bound polynomial in the class number $h\_K$ and logarithmic in $|\mathrm{Discriminant}(K)|$, but I am sure we can get rid of the $h\_K$, or at least change it for a $\log h\_K$, but I'm afraid this would require analytic methods...



Small generators of the ideal class group
First, do not forget that Bach's result is conditional to GRH.


What Bach proves is that for every nontrivial character $\chi$ of the ray class group of $K$ of conductor $\mathfrak{f}$, there is a prime $\mathfrak{p}$ of norm less or equal to $3\log(\Delta\_K^2N\mathfrak{f})^2$ such that $\chi(\mathfrak{p})\neq 1$. In particular, by taking $\mathfrak{f}$ the different of $K$, you get the result you want with a bound $27\log(\Delta\_K)^2$.


