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Let $K$ be a number field and $S$ be a finite set of non-archimedean places of $K$. Let $n>1$ be a natural number.

Question. Does there exist a normal extension $L/K$ of degree $n$ such that $L\otimes_K K_v$ is a field for each $v\in S$ ?

Here $K_v$ denotes the completion of $K$ at $v$.

I know how to construct a not necessarily normal extension. I construct an extension $K'_v/K_v$ of degree $n$ for each $v\in S$, choose a primitive element $\alpha_v\in K_v'$, and write the irreducible polynomial $P_v(X)={\rm Irr}(\alpha_v, K_v', X)$. Then I approximate the constructed polynomials $P_v(X)\in K_v[X]$ by a polynomial $P(X)\in K[X]$ and define $L=K[X]/(P)$. Can this construction be modified so that it would give a normal extension $L/K$?

Alternatively, can one use class field theory to construct an abelian (hence, normal) extension $L/K$ with required properties?

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    $\begingroup$ Unless $8|n$ and $S$ contains a prime extending $2$, it works with cyclic extensions by Grunwald-Wang. $\endgroup$
    – Joachim König
    Commented Nov 24, 2023 at 16:31
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    $\begingroup$ "How can one construct": are you asking whether the statement is true, or are you asking about an explicit construction of a fact you know to hold true? in the first case, I would rephrase. $\endgroup$
    – YCor
    Commented Nov 24, 2023 at 16:57
  • $\begingroup$ Thank you, @JoachimKönig, but I need the general case. $\endgroup$
    – Mikhail Borovoi
    Commented Nov 24, 2023 at 18:37
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    $\begingroup$ Yes (with $L/K$ cyclic) by Artin-Tate Ch X, Thm 6. $\endgroup$
    – user491858
    Commented Nov 25, 2023 at 1:20
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    $\begingroup$ Thank you, @user491858. This is exactly what I need! $\endgroup$
    – Mikhail Borovoi
    Commented Nov 25, 2023 at 6:33

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