# On Grunwald-Wang theorem

Consider (roughtly speaking) the following statements (the Grunwald-Wang theorem)

Theorem 1 (see here for details Wiki): Let $K$ be a number field and $x \in K$. Then under some conditions : $x$ is an $n$-th power iff $x$ is an $n$-th power almost locally everywhere.

Theorem 2 (see theorem (9.2.8) on p541) : Let $K$ be number field and family $(L_p/K_p)_p$ of local abelian extensions. Then under some conditions : there exists an extension $M/K$ such that $M_p \simeq L_p$.

Question 1 : Why are these two theorems equivalent ?

According to Wiki, the fact that $16$ is an 8 power almost locally everywhere but not in $\mathbb{Q}$, implies that there is no cyclic extension of degree $8$ where $2$ is inert.

Question 2 : How to prove this implication ?

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