\part{$p$-adic numbers}


<h3 class=dred>Hensel's lemma</h3>
<p>Assume we know the answer \(x_{k}\) to some \(f(x_{k}) \equiv 0 \mod p^k\), if \( \gcd (p,f'(x_{k})) =1 \implies \exists x_{k+1} = p^{k+1}m + a : f(x_{k+1}) \equiv 0 \mod p^{k+1}\) and \( \frac{f(x_{k})}{p^k}+ mf'(x_{k})  \equiv 0 \mod p\)</p>

