Introduction
相信大家或多或少听过或者熟悉计算机里的（二进制）“补码”这个概念：比如说当我们用$(0001)_2$来表示$1$时，可以用$(1111)_2$ （$= (1111)_2 \text{ xor } (0001)_2 + 1$）来表示$-1$。
而当我们将其转化成我们熟悉的十进制（$(1111)_2 = \sum^{3}_{i=0}1\cdot 2^n = 15$）的话，我们可以观察到下面这几个等式：
而这个4刚好是我们的binary code length。如果我们现在考虑一个无限长的code，那么该如果表达$-1$呢？很显然是$(\cdots1111)_2$对吧，因为它满足
P进数 P-adic Numbers
受到Introduction里的例子的启发，我们做以下定义：
Definition:
Let $p$ be a prime number. A p-adic integer is a formal power series
with $\ a_i \in \{0,1,…,p-1\}$ for all $i \in \mathbb{N}$. The set of all p-adic integers is written as $\mathbb{Z}_p$.
Furthermore, let $f = \frac{g}{h} \in \mathbb{Z}_p$, the denominator of which is not divisible by $p$, then a p-adic integer $\sum^\infty_{i=0}a_i p^i \in \mathbb{Z}_p$ is called the \textbf{p-adic expansion} of $f$ if
and we write
Definition:
Let $p$ be a prime number, $m \in \mathbb{Z}$ an integer. A series of the form
where $a_i \in \{0,…,p-1\}$ for all $i$, is called a p-adic number and the set of all p-adic numbers is written as $\mathbb{Q}_p$.
Furthermore, let $f = p^{m}\frac{g}{h} \in \mathbb{Q}$, a $p$-adic number $\sum^\infty_{i=m}a_i p^i \in \mathbb{Q}_p$ is called the \textbf{p-adic expansion} of $f \in \mathbb{Q}$, if $\sum^\infty_{i=0}a_{i+m} p^i$ is the p-adic expansion of $\frac{g}{h}$.
Absolute values and completeness
Definition: Let $a=\sum_{i=m}^\infty a_ip^i \in \mathbb{Q}_p$. Then we define the p-adic absolute value as follows:
Hensel’s lemma
Theorem:
Let $q \in \mathbb{R}_{>1}$, $K$ be a field with a discrete valuation such that $K$ is complete with respect to the induced $q$-exponential absolute value. Let $\mathcal{O}$ be its discrete valuation ring with the maximal ideal $\mathfrak{p}$ and $\kappa = \mathcal{O}/\mathfrak{p}$.
If $f \in \mathcal{O} [X]$ is a primitive polynomial and there are coprime polynomials $\bar{g}, \bar{h} \in \kappa[X]$ such that
then there exist $g,h \in \mathcal{O}[X]$ with deg$(g) = \text{deg}(\bar{g})$ such that
