On Grunwald-Wang theorem

Solution 1:

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