# every ordered ﬁeld $K$ has a natural valuation $v$, whose residue ﬁeld is an archimedean ordered ﬁeld.

Natural valuation has a convex valuation ring, in fact the valuation ring is the convex hull of $$\mathbb{Z}$$.

How to prove that every ordered ﬁeld $$K$$ has a natural valuation $$v$$, whose residue ﬁeld is an archimedean ordered ﬁeld.

• Well you can first try to prove that the convex hull of $\mathbb{Z}$ in $K$ is a valuation ring of $K$. – nombre Dec 2 '18 at 13:40
• There is a theorem that gives that result right away. Convex hull of any subring in ordred field K is valuation ring. Therefore this one too. – XYZ Dec 2 '18 at 14:03
• Okay, then do you know how the order on this residue field is defined? If so, given $x \in \operatorname{Hull}(\mathbb{Z})$, can you find $n \in \mathbb{N}$ with $\overline{x} \leq \overline{n}$? (where $\overline{a}= a+\mathfrak{m}$ and $\mathfrak{m}$ is the maximal ideal of the valuation ring) – nombre Dec 2 '18 at 15:28
• I know that $P=\{\bar a : a\in \mathbb{Z} , a\geq 0 \}$ is an ordering in residue field. How to use it to find $n$? – XYZ Dec 2 '18 at 16:18
• Correction: $a$ in ordering P is in convex hull of $\mathbb{Z}$ – XYZ Dec 2 '18 at 16:29