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.
Finite extension of $\mathbb Q_p$
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number-theory
field-theory
finite-fields
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2This looks like homework. What have you tried? – 2011-05-14
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2Please 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. – 2011-05-14