Finite extension of $\mathbb Q_p$

Question

Let $\mathbb K/\mathbb Q_p$ be a finite extension of $p-$adic field $\mathbb Q_p$. Let ${\mathcal O}=\{x\in K\;:\;|x|\leq1\}$ and ${\mathcal P}=\{x\in K:\;|x|<1\}$, here $|\cdot|$ is the absolute value. Show that the quotient ring $\mathcal O/\mathcal P$ is a finite field. What is the cardinal of its and show a complete system of representatives of the residue classes of this quotient ring.

Answer 1

${\cal{O}}$ is a very distinguished ring inside $\mathbf{K}$...what is it?

Try studying the situation first when $\mathbf{K}=\mathbf{Q}_p$.