p-adic numbers

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evinda
Posts: 2
Joined: Tue Oct 14, 2014 2:33 pm

p-adic numbers

Post by evinda » Tue Oct 14, 2014 2:35 pm

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? :?

Pengolodh
Posts: 6
Joined: Wed Nov 21, 2012 9:03 am

Re: p-adic numbers

Post by Pengolodh » Wed Oct 15, 2014 12:11 pm

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.

evinda
Posts: 2
Joined: Tue Oct 14, 2014 2:33 pm

Re: p-adic numbers

Post by evinda » Wed Oct 15, 2014 5:21 pm

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.
I want to show that $\frac{1}{2}$ is an integer $5$-adic. How can I do this? :?

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 $i\ge 1$

? :?

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