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Trying to get my head around p-adics,as i learn some more advanced number theory techniques but im stuck on an exercise.

If we let $\alpha = a_0 + a_1p + a_2p^2 + \dots = \sum_{n=0}^\infty a_np^n$ be the expansion of a p-adic unit, then show that $\beta = -\alpha$ has expanson $\beta = \sum_{n=0}^\infty b_np^n$ with $b_0 = p-a_0$ and $b_n = p-1-a_n$ for $n \geq 1$.

I then have to go onto to determine the first four digits of $\frac{1}{\alpha}$ where $\alpha = 2 + 1\cdot 5 + 3 \cdot 5^2 + 2 \cdot 5^3 \dots \in \mathbb{Z}_5$ (expressing it similarly as a 5-adic expansion).

I'm new to p-adic's as a concept. For the first question i tried to equate coefficients and then deduce each coefficient but i cannot seem to arrive at the correct $b_n$.

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  • $\begingroup$ Try calculating $\beta+\alpha$. All going well, you should get 0 $\endgroup$ – Mathmo123 Mar 6 '15 at 0:24
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I still think the best way to think of $p$-adic integers, and the best way to do hand computations, is to use $p$-ary notation: binary if $p=2$, ternary if $p=3$, etc. Then the $5$-ary expansion of your number $\alpha$ is, rather, $$ \dots2312; $$ Notice that without the dots, this is just the $5$-ary expansion of the positive integer whose decimal expansion is $332$. The advantage of writing $p$-adic numbers this way is that to add, subtract, and multiply them, you use exactly the algorithms you learned in elementary school. The carries proceed right to left, same as in school, and all you need to have in front of you is a $p$-ary multiplication table and maybe also an addition table.

So, for instance, to calculate $-\alpha$, you put it underneath $\dots00000;$ and do your subtractions. First, you get $3;\,$, but you need a borrow and carry, and proceeding in the usual way, you find that $-\alpha$ has expansion $\dots2133;\,$. Alternatively, just as you may make change for $\$100$ by subtracting the price from $\$99.99$ and then adding a penny to the result, you may find $-\alpha$ by subtracting from $\dots4444444;\,$ and adding $\dots00001;$ to the result.

Division is a little trickier, since the elementary-school algorithm proceeds left to right, while $p$-adically, you must work right to left. If I were handier with LaTeX, I’d show you how to calculate $1/\alpha$, but I’ll have to leave it to you. (In case you’re still stumped, I can go into greater detail in a messy display.)

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