The Field of $p$-adic Numbers

The ring $\mathbf{Q}_{10}$ of $10$-adic numbers is isomorphic to $\mathbf{Q}_{\hspace{.2ex}2}\times \mathbf{Q}_{\hspace{.2ex}5}$ (see Exercise 58), so it is not a field. For example, the element $\ldots8212890625$ corresponding to $(1,0)$ under this isomorphism has no inverse. (To compute $n$ digits of $(1,0)$ use the Chinese remainder theorem to find a number that is $1$ modulo $2^{n}$ and 0 modulo $5^n$.)

If $p$ is prime then $\mathbf{Q}_p$ is a field (see Exercise 57). Since $p\neq 10$ it is a little more complicated to write $p$-adic numbers down. People typically write $p$-adic numbers in the form

$$\frac{a_{-\!d}}{p^{d}} + \cdots + \frac{a_{-1}}{p} + a_0 + a_1 p + a_2 p^2 + a_3 p^3 + \cdots$$

where $0\leq a_i < p$ for each $i$.