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If $p$ is a prime number, then every rational number $r$ can be written in the form $r=p^j \frac{a}{b}$ where $\gcd(p, ab)=1$. We define $|r|_p = p^{-j}$, and define a metric on $\Q$ by $d(u,v) = |u-v|_p$; in other words, $u$ and $v$ are close to each other if they are congruent modulo a larger power of $p$.

The completion of $\Q$ with respect to this metric is $\Q_p$, which is itself a field in the natural way. Any non-zero element $x \in \Q_p$ can be written uniquely as the sum of a convergent series $x = \sum_{n=n_0}^{+\infty} a_n p^n$, where $a_n \in \{ 0, \cdots, p-1 \}$, $n_0 \in \Z$ is such that $|x|_p=p^{-n_0}$, and $a_{n_0} \neq 0$.

Example: the element $a = 3 + 7 + 2 \cdot 7^2 + 6 \cdot 7^3 + 7^4 + 2 \cdot 7^5 + \cdots$ satisfies $a^2=2$, since truncating the expansion after $n$ terms and squaring yields a number which is congruent to $2 \bmod 7^n$.

Knowl status:
  • Review status: reviewed
  • Last edited by John Cremona on 2018-05-23 14:42:00
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