TITLE: Does every ideal class contains a prime ideal that splits?
QUESTION [11 upvotes]: 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.

REPLY [6 votes]: The answer to both A and B is no.  Take the non-Galois field $L = \mathbb{Q} (\sqrt[4]{-5})$, and let $N = L(i)$, the Galois closure of $L$.  Computing with PARI/GP tells me $L$ has cyclic class group of order $4$ and $N$ has class group of order $2$.  Thus, the map $N \colon \mathrm{Cl}_N \rightarrow \mathrm{Cl}_L$ induced by the relative norm of ideals is not surjective. 
Assume, to reach a contradiction, that one of the classes of order $4$ in $\mathrm{Cl}_L$ contains a prime $\mathfrak{p}$ lying over a prime $p \in \mathbb{Z}$ that is completely split in $L$.  It is well-known that this happens if and only if $p$ is completely split in $N$. Then $\mathfrak{p}$ is the norm of an ideal $\mathfrak{P}$ in $N$.  But then the class of $\mathfrak{p}$ is the norm of the class of $\mathfrak{P}$, contradicting the fact that the norm map on classes is not surjective.

A is true if we make the additional assumption that the normal closure of $L$ is linearly disjoint over $L$ from the Hilbert class field of $L$.
To see this, let $N$ be the normal closure of $L$ over $\mathbb{Q}$, let $H_N$ be the Hilbert class field of $N$, and let $H_L$ be the Hilbert class field of $L$.  We must show that for each $\sigma \in G(H_L/L)$, there exists some prime ideal of $L$, completely split over $\mathbb{Q}$ and whose Frobenius for $H_L/L$ is $\sigma$.  
We need some facts:

$H_N$ is Galois over $\mathbb{Q}$
$N H_L \subset H_N$

For 1, consider an embedding $\tau \colon H_N \hookrightarrow \overline{\mathbb{Q}}$. We have $\tau (N) = N$, so $\tau (H_N)$ is an abelian unramified extension of $N$, hence is contained in $H_N$  It follows that $\tau(H_N) = H_N$ and 1 follows.
For 2, consider a prime ideal $\tilde{\mathfrak{P}}$ in $N H_L$, which is Galois over $L$. Because $H_L/L$ is unramified, the inertia group for $\tilde{\mathfrak{P}}$ over $L$ is contained in $G(N H_L /H_L)$.  But the definition of inertia groups also shows that the inertia group for $\tilde{\mathfrak{P}}$ over $N$ is contained in that for $\tilde{\mathfrak{P}}$ over $L$.  Thus, the inertia group for $\tilde{\mathfrak{P}}$ over $N$ consists only of automorphisms that fix $N$ and $H_L$, so it is trivial and $N H_L/N$ is unramified. Since $G(N H_L/N)$ injects into $G(H_L/L)$ by restriction, $N H_L/N$ is abelian and 2 follows.
Now let us return to considering our $\sigma$ in $G(H_L/L)$.  By 1 and the assumption that $N$ and $H_L$ are linearly disjoint over $L$, we can lift $\sigma$ to an automorphism $\tilde{\sigma}$ in $G(H_N/N)$.  By Chebotarev's density theorem, we can find some prime ideal $\mathfrak{P}$ in $H_N$, unramified over $\mathbb{Q}$ and whose Frobenius over $\mathbb{Q}$ is $\tilde{\sigma}$.  The decomposition group of $\mathfrak{P}$ over $\mathbb{Q}$ is thus contained in $G(H_L/L)$, so the prime $p = \mathfrak{P} \cap \mathbb{Z}$ is completely split in $L$.  
Let $\mathfrak{p} = \mathfrak{P} \cap N$. By the formalism of the Artin symbol, 
 the Frobenius for $\mathfrak{P}$ over $N$ is $\tilde{\sigma}$ and its restriction to $H_L$, namely $\sigma$ is itself the image under the Artin map of the ideal $N_{N/L} \mathfrak{p}$.  But again, because $p$ is completely split up to $L$, it is also completely split up to $N$ and so $N_{N/L} \mathfrak{p}$ is a prime ideal of $L$ with degree $1$ and Frobenius equal to $\sigma$ as desired.