$\dim(H^1(G_K,\mathbb{Q}_p(1)))=1+[K:\mathbb{Q}_p]$ when $\ell=p$ and $1$ otherwise? Let $K$ be a finite extension of $\Bbb{Q}_{\ell}$ and let $G_K=\mathrm{Gal}(\overline{K}/K)$. Then is the following true, by the Euler-Poincare characteristic formula?
$\dim(H^1(G_K,\mathbb{Q}_p(1)))=1+[K:\mathbb{Q}_p]$ when $\ell=p$ and $1$ when $\ell\neq p$.
$\mathbb{Q}_p(1)$ is the $p$-adic cyclotomic character.
 A: The Euler-Poincare characteristic formula (due to John Tate) for local Galois Cohomology says that
Let $K$ be a finite extension of $\Bbb{Q}_{\ell}$ for any prime $\ell$. Let $G_K$ be its absolute Galois group. Then we have the following
$$\dim(H^0(G_K,V))-\dim(H^1(G_K,V))+\dim(H^2(G_K,V))=0$$
if $\ell\neq p$ and $-[K:\Bbb{Q}_p]\dim(V)$ when $\ell=p$.
Here $V$ is $\Bbb{Q}_p(1)$ and it's one dimensional. The Euler-Poincare characteristic can be interpreted like this: The dimension of $H^0(G_K,V)$ is $\dim(V^{G_K})$ which counts the multiplicity of the trivial representation $\Bbb{Q}_p$ as a subrepresentation of $V$. By duality, $H^2(G_K,V)\cong H^0(G_K,V^*(1))$. Hence $\dim(H^2(G_K,V^*(1)))$ is the multiplicity of the $p$-adic cyclotomic character $\Bbb{Q}_p(1)$ as a quotient of $V$.
Since $\Bbb{Q}_p(1)$ is an one dimensional nontrivial representation and hence it is irreducible and hence $\Bbb{Q}_p$ (the trivial representation) does not occur as a subrepresentation of it. Thus $\dim(H^0(G_K,\Bbb{Q}_p(1)))=0$. Also, $\Bbb{Q}_p(1)$ is a quotient of itself (quotient via the trivial subspace). Hence $\dim(H^2(G_K,\Bbb{Q}_p(1)))=1$. Thus, by the Euler-Poincare characteristic formula,
\begin{gather*}
    \dim(H^1(G_K,\Bbb{Q}_p(1)))=0+1+[K:\Bbb{Q}_p]\dim(\Bbb{Q}_p(1))=\begin{cases}[K:\Bbb{Q}_p]+1\quad\ell=p\\1\qquad\qquad\qquad\ell\neq p\end{cases}
\end{gather*}
