Example 3

On a division ring $K$, a real valuation is a function $v$ defined on $K^*$ with values in $\mathbf{R}$ which satisfies the following conditions: a) if $x \in K^*$ and $y \in K^*$, then $v(xy) = v(x) + v(y)$; b) if in addition $x + y \neq 0$, then $v(x + y) \geq \inf(v(x), v(y))$. If $a$ is any real number $> 1$, we can then define an absolute value on $K$ by putting $|x| = a^{-v(x)}$ for $x \neq 0$, and $|0| = 0$. For the relation $v(xy) = v(x) + v(y)$ for $x \neq 0$ and $y \neq 0$ implies the relation $|xy| = |x|.|y|$ for these values of $x$ and $y$, and this relation is trivially true if one of $x, y$ is zero; likewise, from the relation $v(x + y) \geq \inf(v(x), v(y))$ for $x \neq 0, y \neq 0$ and $x + y \neq 0$ we deduce $|x + y| \leq \sup(|x|, |y|) \leq |x| + |y|$, and these inequalities are still satisfied if one of $x, y, x + y$ is zero. In particular, if $v_p(x)$ is the $p$-adic valuation on the field $\mathbf{Q}$ of rational numbers (the exponent of $p$ in the decomposition of $x$ into a product of prime factors), then the corresponding absolute value $|x|_p = p^{-v_p(x)}$ is called the $p$-adic absolute value on the field $\mathbf{Q}$ (cf. Chapter III, § 6, Exercise 23).