I am reading Serre's Galois Cohomology and am stuck trying to understand a statement made on page 95, II.5.5 Proposition 18.
Let $k$ be a $p$-adic field and $A$ be a finite unramified $G_k$-module. There is a claim that $H^i(k_{nr}/k, A)$ has order equal to that of $H^0(k,A)$.
The justification is that $H^0(\hat{\mathbb{Z}},A)$ and $H^1(\hat{\mathbb{Z}},A)$ have the same number of elements. I don't see how this statement is true.