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Let $p\neq 2$ be prime. I am asked in a revision question to find the $p$-adic expansion of $(1+2p)/(p-p^3)$. The best I could do was find the $p$-adic norm, which I got as $p$ (please correct me if this is wrong) thus giving the expansion as $$ \frac{1+2p}{p-p^3} = a_{-1}p^{-1} + a_0 + a_1 p + \dots $$ where $a_i \in \{0,\dots,p-1\}$. I've tried several other things and none of them seem to be fruitful. No questions of this type were even addressed in lectures... Can anyone help?

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up vote 10 down vote accepted

Write it as $$\frac{1+2p}{p}\cdot\frac{1}{1-p^2}.$$ Because $|p^2|_p=\frac{1}{p^2}<1$, we can use the geometric series to give the $p$-adic expansion of $\frac{1}{1-p^2}$. Then shift it down by one due to the $\frac{1}{p}$, then multiply by $1+2p$ (the action of this on the coefficients is not too hard to figure out).

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+1 Brilliant! Very slick. – Fahad Sperinck Jun 11 '11 at 12:13

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