Let $K$ be a number field, $d$ a positive integer and $S$ a finite set of places of $K$.

By Cebotarev, there exists a finite set of finite places $T$ disjoint from $S$ such that the conjugacy classes of geometric Frobeni $F_v$ ($v\in T$) fill up $\mathrm{Gal}(K^\prime/K)$ for any $K^\prime/K$ Galois of degree at most $d$.

For a finite field extension $L/K$, let $T_L$ be the set of places of $L$ lying over $T$. We use similar notation for $S_L$.

Question. Let $L/K$ be a finite extension, not necessarily Galois. Then $T_L$ is disjoint from $S_L$. Do the conjugacy classes of geometric Frobeni $F_w$ ($w\in T_v$) fill up $\mathrm{Gal}(L^\prime/L)$ for any $L^\prime/L$ Galois of degree at most $d$?

Let $K$ be a number field, $d$ a positive integer and $S$ a finite set of places of $K$.

By Cebotarev, there exists a finite set of finite places $T$ disjoint from $S$ such that the conjugacy classes of geometric Frobeni $F_v$ ($v\in T$) fill up $\mathrm{Gal}(K^\prime/K)$ for any $K^\prime/K$ Galois of degree at most $d$ (Edit) and unramified outside $S$.

For a finite field extension $L/K$, let $T_L$ be the set of places of $L$ lying over $T$. We use similar notation for $S_L$.

Question. Let $L/K$ be a finite extension, not necessarily Galois. Then $T_L$ is disjoint from $S_L$. Do the conjugacy classes of geometric Frobeni $F_w$ ($w\in T_v$) fill up $\mathrm{Gal}(L^\prime/L)$ for any $L^\prime/L$ Galois of degree at most $d$ (Edit) and unramified outside $S_L$?