Proposition 4

Let $C$ be an integral domain, $K$ its field of fractions, $C^* = C - \{0\}$ and $v : C \to \Gamma_\infty$ a valuation on $C$. Then there exists a unique valuation $w$ on $K$ which extends $v$ and $w(K^*)$ is the subgroup of $\Gamma$ generated by $v(C^*)$.

By Theorem 2 of Algebra, Chapter I, § 2, no. 7, there exists a unique homomorphism $w$ from $K^*$ to $\Gamma$ which extends $v|C^*$ and $w(K^*)$ is generated by $v(C^*)$. It remains to prove that $w$ satisfies axiom (VL$_{\text{II}}$). Then let $x' \in K^*, y \in K^*$ be such that $x + y \in K^*$; there exists $a \in \mathbf{C}^*$ such that $ax \in \mathbf{C}^*$ and $ay \in \mathbf{C}^*$, whence $a(x + y) \in \mathbf{C}^*$. Since the restriction of $w$ to $\mathbf{C}^*$ satisfies (VL$_{\text{II}}$),

Eliminating $w(a)$ from both sides, we obtain