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Let $\mathcal{P}$ a prime ideal in $\mathcal{O}_K$. Show that $a^{N(\mathcal{P})} \equiv a \pmod{\mathcal{P}}$, $a \in \mathcal{O}_K$.

I have this: $N(\mathcal{P})=p^f$, $f$ is the inertia degree of $\mathcal{P}$ above $p$.

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You are posting several short questions with no comment from you about background context, what you have tried, and so on. I am getting the sense that you're posting questions from an exam for us to solve for you. What exactly is prompting this list of algebraic number theory questions from you this evening? –  KCd Dec 19 '12 at 3:49
@KCd You Right, I have an exam in a few days. My apologies for causing inconvenience. –  P. M. O. Dec 19 '12 at 3:55
@KCd This is the last of the day. Thanks! –  P. M. O. Dec 19 '12 at 3:59
What you are posting in this question includes Fermat's little theorem as a special case: $a^p \equiv a \bmod p$ for any integer $a$, when $p$ is a prime number. Can you prove that? If so, does that suggest how to prove the result in your subject line? –  KCd Dec 19 '12 at 4:02

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