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Suppose you have a number field $L$, and a non-zero ideal $I$ of the ring of integers $O$ of $L$.

Question part A: Is there prime ideal $\mathcal{P} \subseteq O$ in the ideal class of $I$ such that $p =\mathcal{P} \cap \mathbb{Z}$ splits completely in $O$?

If the extension $L/\mathbb{Q}$ is Galois I think one can show that the answer is yes. In fact I think it is possible to show that for every $I$ there are infinitely many such $\mathcal{P}$'s. The necessary tools for this are class field theory and Chebotarev's density theorem. On the other hand I'm not asking for infinitely many primes, only for one. So to be concrete:

Question part B: If the answer to A is yes, is it possible to give a proof that avoids showing that there are infinitely many $\mathcal{P}$'s?

Thank you.

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    See also Narkiewicz, _Elementary and Analytic Theory of Algebraic Numbers_, corollary 7 from §7.2, page 347 (notations on page 93).2018-12-02

2 Answers 2