# p-adic分析note| Solving Congruences Modulo $p^n$

## Solving Congruences Modulo $p^n$

The “p-adic numbers” we have just constructed are closely related to the problem of solving congruences modulo powers of $p$. We will look at some examples of this.

Let’s start with the easiest possible case, an equation which has solutions in $\mathbb{Q}$, such as
$$X^{2}=25 .$$
We want to consider it modulo $p^{n}$ for every $n$, i.e., to solve the congruences
$$X^{2} \equiv 25 \quad\left(\bmod p^{n}\right) .$$
Now, of course, our equation has solutions already in the integers: $X=$ $\pm 5$. This automatically gives solutions of the congruence for every $n$; just put $X \equiv \pm 5\left(\bmod p^{n}\right)$ for every $n$.

## 解同余模 $p^{n}$

$$X^{2}=25 \text {. }$$

$$X^{2} \equiv 25 \quad\left(\bmod p^{n}\right) .$$

p-adic分析note| Solving Congruences Modulo $p^n$ 请认准Calculus-do™. Calculus-do™为您的留学生涯保驾护航。