Hello!!!!

I want to show that 1/2 is an integer p-adic and to find the first five terms of its powerseries.

Firstly, how do we conclude that 1/2 is an integer p-adic number?

## p-adic numbers

### Re: p-adic numbers

Well, it isn't an integer for $p=2$. For an odd prime $p$, $1/2=\sum^\infty_{i=0}a_i\,p^i,$ where $a_0=(p+1)/2$ and $a_i=(p-1)/2$ for $i\ge 1$. Just use the geometric series for $1/(1-p)$, multiply by $(p-1)/2$ and add 1.

### Re: p-adic numbers

I want to show that $\frac{1}{2}$ is an integer $5$-adic. How can I do this?Pengolodh wrote:Well, it isn't an integer for $p=2$. For an odd prime $p$, $1/2=\sum^\infty_{i=0}a_i\,p^i,$ where $a_0=(p+1)/2$ and $a_i=(p-1)/2$ for $i\ge 1$. Just use the geometric series for $1/(1-p)$, multiply by $(p-1)/2$ and add 1.

How do we know that:

$1/2=\sum^\infty_{i=0}a_i\,p^i,$ where $a_0=(p+1)/2$ and $a_i=(p-1)/2$ for $i\ge 1$

?