Let $F$ be a finite field extension of $\mathbb Q$, with ring of integers $\mathcal O\_F$. Let $G=CL(\mathcal O\_F)$ be the class group of $\mathcal O\_F$ and let $M$ be a minimal set of generators of $G$ (i.e. a subset of $G$ of minimal cardinality which generates $G$).


Question: a) Can one find a set $M$ such that any element in $M$ has a prime ideal (of $\mathcal O\_F$) representative?


b) How big is the cardinality of $M$ in terms of the the cardinality of $G$?


c) If the answer to a) is yes. Is there a simple proof or a suitable reference?


d) If the answer to a) is no. Is there an explicit counter example?



Generators of class groups
The group $Cl(O\_F)$ is the Galois group of an abelian extension of $F$ (the Hilbert class field of $F$). Every element of the Galois group of a finite Galois extension $E/F$ is represented by a prime ideal of $F$ (the Cebotarev density theorem). So the answer to (a) is yes. This is proved in Serge Lang's book on algebraic number theory. 


The answer to (b) is obviously less than the cardinality of $G$ (it seems to me this is a question of finite abelian groups). 


[You do not need Cebotarev but only that $L(1,\chi)\neq 0$ for Hecke characters associated to $F$. All this is worked out in any book on class field theory]. 


