TITLE: Chebotarëv data over number fields
QUESTION [9 upvotes]: Define a Chebotarëv datum over a number field $K$ to be a finite group $G$ together with a map $\mathfrak{p}\mapsto\gamma_{\mathfrak{p}}$ from a cofinite set of primes of $K$ into the set of conjugacy classes of $G$ such that for every conjugacy class $c\subset G$, the proportion of $\mathfrak{p}$ with $\gamma_{\mathfrak{p}}=c$ is $\operatorname{Card}c/\operatorname{Card}G$.  
Two Chebotarëv data $(G,\gamma)$, $(G',\gamma')$ over the same number field are said to be equivalent if there is an isomorphism $\varphi:G\to G'$ such that $\varphi(\gamma_{\mathfrak{p}})=\gamma'_{\mathfrak{p}}$ for almost all $\mathfrak{p}$.  If so, we identify the two.
Every finite galoisian extension $L$ of $K$ gives rise to a Chebotarëv datum $(\operatorname{Gal}(L|K),\gamma_{L|K})$ (Chebotarëv's density theorem).
Moreover, if $L_1$ and $L_2$ are two finite galoisian extensions of $K$ for which the associated Chebotarëv data $(\operatorname{Gal}(L_i|K),\gamma_{L_i|K})$ are equivalent, then $L_1=L_2$ (see Lemma 1, p. 174, of Mazur's recent article).
Question.  Does every Chebotarëv datum over a given number field $K$ arise from some finite galoisian extension $L$ of $K$ ?

REPLY [8 votes]: In addition to David's answer, one can provide explicit counterexamples, at least in specific cases.
For simplicity, take $K=\mathbf{Q}$ and $G=\{\pm 1\}$. Define the Chebotarëv datum $\gamma$ by $\gamma(p) = -1$ if $p \equiv 1 \pmod{4}$ and $\gamma(p)=1$ if $p \equiv 3 \pmod{4}$. If $\gamma$ comes from a quadratic field $L$ of discriminant $D$, then there exists a quadratic character $\chi : (\mathbf{Z}/D\mathbf{Z})^* \to \{\pm 1\}$ such that $\gamma(p) = \chi(p)$ for almost all primes $p$. We now have a contradiction by looking at primes $p \equiv 1 \pmod{4D}$ (there are infinitely many such primes).
In fact, by Dirichlet's theorem on primes in arithmetic progressions, the Chebotarëv datum $\gamma$ cannot be modified on a subset of natural density zero in order to come from a quadratic field.