discrete valuation

《Local Fields》 · 第 5 页 · Chapter I，Section 1「Definition of Discrete Valuation Ring」

If $x\ne0$ is any element of $A$, one can write $x=\pi^n u$, with $n\in\mathbb N$ and $u$ invertible; the integer $n$ is called the valuation (or the order) of $x$ and is denoted $v(x)$; it does not depend on the choice of $\pi$. Let $K$ be the field of fractions of $A$, $K^*$ the multiplicative group of non-zero elements of $K$. If $x=a/b$ is any element of $K^*$, one can again write $x$ in the form $\pi^n u$, with $n\in\mathbb Z$ this time, and set $v(x)=n$. The following properties are easily verified: a) The map $v:K^*\to\mathbb Z$ is a surjective homomorphism. b) One has $v(x+y)\geq\operatorname{Inf}(v(x),v(y))$. (We make the convention that $v(0)=+\infty$.)