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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.

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This looks like homework. What have you tried? – Soarer May 14 '11 at 3:03
Please don't post in the imperative mode ("Show", "prove", "construct"). You aren't giving us an assignment, you are, I think, trying to ask a question. So ask, don't tell. – Arturo Magidin May 14 '11 at 3:22

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

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

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