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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?

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up vote 3 down vote accepted

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].

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A sharp answer for part $b$ is that $|M|\leq \log_2(|G|)$. –  Kevin Ventullo Apr 30 '13 at 22:18
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Yes, indeed. the equality is attained when $G$ is abelian of exponent two. –  Aakumadula May 1 '13 at 2:39
    
Dear Aakumadula, thanks a lot for your fantastic answer. If I understand you correctly, then your argument shows in addition that any class of $G$ ((not only of $M$)) can be represented by a prime ideal? –  Guillaume Pastorini May 1 '13 at 11:41
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Dear Guillaume, not at all! Yes, indeed, any class of $G$ can be represented by a prime ideal. This is analogous to Dirichlet's theorem on infinitude of primes in arithmetic progressions; indeed, that any element of $({\mathbb Z}/m{\mathbb Z})^*$ (* denotes the group of units) is represented by primes may be interpreted as saying that elements of certain generalized ideal classes are represented by primes. –  Aakumadula May 1 '13 at 12:24
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