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Constructing a cyclic extension $L$ with given local behavior of a global field $K$ such that $L$ is normal over a subfield $F$ of $K$

Let $F$ be a global field without real places (that is, a function field or a totally imaginary number field). Let $K/F$ be a cyclic extension of degree $n$. Let $S$ be a ${\rm Gal}(K/F)$-invariant ...