Can anyone teach me about p-adic expansion? especially the case where we have to expand a square root.
I need to know how to construct them. for example: the 7-adic expansion of $\sqrt{305}$.
This is a general question and not an assignment or anything, so i cannot post the progress.
That is just a random example, please use any other example to explain if it's easier to understand.
edit: I'm stuck with the cases where i have to expand the square roots, expanding rational numbers is fine though
Best Answer
The $p$-adic number will be ....dcba
Start with units, $a^2=305=4 \bmod 7$, so $a=2$.
Next, the sevens $(7b+2)^2=305=11\bmod 49$, so $28b=7\bmod 49$, and $b=2$.
Next, the 49s: $(49c+7*2+2)^2=305\bmod 343$, and so on.
Although it wasn't guaranteed that $a$ would exist, because $305$ might not have been a quadratic residue $\bmod 7$, all the other digits must exist.