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?