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Theorem 4 of Eric Bach's "Explicit bounds for primality testing and related problems" states the following:

Let $K$ be a number field of degree greater than 1. Let $d$ be the absolute value of the discriminant of $K$. Let $\chi$ be a nonprincipal character of the ideals of $k$ that is defined modulo $\mathfrak{f}$. Then: \begin{align*} \chi (p) &\neq 1 \text{ occurs for } N(\mathfrak{p}) \leq 3\log^{2}(d^2N(\mathfrak{f})) \\ \chi (p) & \neq 0,1 \text{ occurs for } N(\mathfrak{p}) \leq 12\log^{2}(d^2N(\mathfrak{f}))\\ \chi (p) & \neq 0,1 \text{ and } \text{degree}(\mathfrak{p})=1 \text{ occurs for } N(\mathfrak{p}) \leq 18\log^{2}(d^2N(\mathfrak{f})) \end{align*}

How do I get the result that the ideal class group is generated by the prime ideals of norm less than $12\log^{2}(d)$ from above theorem?

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Let $G$ be the ideal class group, and let $H$ be the subgroup generated by the prime ideals of norm at most $3\log^2(d^2)=12\log^2(d)$. Assume that $H$ is a proper subgroup of $G$. Then there is a nontrivial character of $G$ that is trivial on $H$. This character is trivial on the prime ideals of norm at most $3\log^2(d^2)=12\log^2(d)$, contradicting the first part of the theorem (with $\mathfrak{f}=\mathfrak{o}$). Hence $H=G$, and we are done.

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  • $\begingroup$ If prime ideals of norm at most 3log^2(d) is enough for generating ideal class group, why do we come across the term 12log^2(d) in the literature? $\endgroup$
    – Rashad Ek
    Commented Feb 20 at 21:57
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    $\begingroup$ @RashadEk I updated my response. The point is that $3\log^2(d^2)$ in Bach's theorem is the same as $12\log^2(d)$. If you like my answer, please accept it officially (so that it turns green). Thanks in advance! $\endgroup$
    – GH from MO
    Commented Feb 21 at 2:33
  • $\begingroup$ Thank you for the answer. I have one more question. Erich Bach defined the characters on the prime ideals of ring of integers. In your proof you used a character on the ideal class group. How do we connect these two characters? $\endgroup$
    – Rashad Ek
    Commented Feb 21 at 15:16
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    $\begingroup$ @RashadEk Eric Bach talks about characters of $\mathrm{Gal}(E/K)$ where $E/K$ is an abelian extension. By class field theory, the Galois groups of abelian extensions of $K$ can be identified (via the Artin reciprocity map) with quotients of ray class groups of $K$. In particular, the Galois group of the Hilbert class field of $K$ is isomorphic to the ideal class group of $K$. So what Eric Bach talks about includes the characters of the ideal class group of $K$. Note that the ideal class group itself is a quotient of the multiplicative group of nonzero fractional ideals of $K$. $\endgroup$
    – GH from MO
    Commented Feb 21 at 16:19

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