$V = H_\infty\otimes_{\mathbb{Z}_p}\mathbb{Q}_p$ is finite-dimensional

Let $p$ be a prime, $F_n = \mathbb{Q}(\mu_{p^{n+1}})$, $A_n$ the $p$-part of the class group of $F_n$, and $H_\infty = \varprojlim_n A_n$ the inverse limit under the norm maps. Then

is a finite-dimensional $\mathbb{Q}_p$-vector space.

This follows from Iwasawa's theorem that $H_\infty$ is a finitely generated torsion module over $\Lambda = \mathbb{Z}_p[[\Gamma]]$; $\dim_{\mathbb{Q}_p} V$ is Iwasawa's $\lambda$-invariant. Finite-dimensionality is what makes "the characteristic polynomial of $\gamma$ acting on $V_i$" in the source meaningful.