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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.20, of Mazur's recent article).

Question. Does every Chebotarëv datum over a given number field $K$ arise from some finite galoisian extension $L$ of $K$ ?

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 $K$ ?

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.20, of Mazur's recent article).

Question. Does every Chebotarëv datum over a given number field $K$ arise from some finite galoisian extension $L$ of $K$ ?

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.20, of Mazur's recent article).

Question. Does every Chebotarëv datum over a given number field $K$ arise from some finite galoisian extension $L$ of $K$ ?

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 $\rm{Card}c/\rm{Card}G$.

Two Chebotarëv data $(G,\gamma)$, $(G',\gamma')$ over the same number field are said to be equivalent if $\gamma_{\mathfrak{p}}=\gamma'_{\mathfrak{p}}$ for almost all $\mathfrak{p}$. If so, we identify the two.

Every finite galoisian extension $L$ of $K$ gives rise to a Chebotarëv datum $(\rm{Gal}(L|K),\gamma_{L|K})$ (Chebotarëv's density theorem).

Moreover, if $L_1$ and $L_2$ are two finite galoisian extensions of $K$ for which the associated Chebotarëv data $(\rm{Gal}(L_i|K),\gamma_{L_i|K})$ are equivalent then $L_1=L_2$ (see Lemma 1, p.20, of Mazur's recent article).

Question. Does every Chebotarëv datum over a given number field $K$ arise from some finite galoisian extension $L$ of $K$ ?

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.20, of Mazur's recent article).

Question. Does every Chebotarëv datum over a given number field $K$ arise from some finite galoisian extension $L$ of $K$ ?