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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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1 Answer 1

From a modern perspective, Theorem 1 is easier, as it follows immediately from the vanishing of $\Sha^1(K,\mu)$ and Kummer theory.

Theorem 2, on the other hand, requires the vanishing of $\mathrm{coker}^1(K,A)$, the cokernel of the localization map, where A is the potential global galois group (in both cases assuming that we are not in a special case). Theorem 2 is closer to the one originally proven by Grunwald (and later corrected by Wang).

Q2: The implication requires class field theory to prove. The point is that the norm residue symbol of $16$ would vanish everywhere except at $2$, contradicting the product formula/global reciprocity law.

Q1: Since Theorem 2 lets us prescribe the ramification at a finite number of places, we can use the same method to conclude Theorem 1 from Theorem 2.

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Note that there's a latex error. –  John Ma Jul 4 at 12:36
Well, is there any way to display an upper case sha? If not, I'd rather leave it as an error. –  user252374 Jul 4 at 13:11
Can you just use \text{Sha} ? –  John Ma Jul 4 at 13:17
why? I want the symbol sha (, and this way anyone reading the post will understand that. –  user252374 Jul 4 at 13:29

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