Definition 1.2 ( $p$ -adic absolute value). Let $K = ℚ$ , and $p$ be a prime. For $0 \neq x \in ℚ$ , write $x = p^{n} \frac{a}{b}$ , where $(a, p) = 1$ , $(b, p) = 1$ . The $p$ -adic absolute value is defined to be

| x |_{p} = {\begin{matrix} 0 & x = 0 \\ p^{- n} & x = p^{n} \frac{a}{b} \end{matrix}