Here a non-archimedean field means a field $k$ whose topology is induced from a non-archimedean norm $| \cdot |: k \to \mathbb{R}_{\geq 0}$. As a reminder, a ring $A$ is *adic* if there is an ideal $I \subset A$, called the ideal of definition, such that $(I^n)_n$ forms a system of neighborhood of zero, and a ring $A$ is called *Huber* if there is an open adic subring $A_0 \subset A$ such that $A_0$ has a finitely generated ideal of definition.

In this case of a non-archidedean field $k$, what would $k_0$ be? I want to guess $k$ or $\mathcal{O}_k$, but can't show either is adic with finitely generated ideal of definition.