Arithmetic in an Algebraic Number Field

Abstract

6.1. Let F be a field. A map $\,\nu : F\to {\bold{R}}\cup\{\infty\}$ is called an order function of F if it satisfies the following conditions:

• (i) $\,\nu(x)=\infty\,\Longleftrightarrow\, x=0;$

• (ii) $\,\nu(xy)=\nu(x) +\nu(y);$

• (iii) $\,\nu(x+y)\ge\text{Min}\big\{\nu(x),\,\nu(y)\big\};$

• (iv) There exists an element $\,z\in F,\,\ne0,\,$ such that $\,\nu(z)\ne 0.$