Fulton, Exercise 2.28: Show that $R$ is a DVR.

Question

We define as order function on a field $K$ a function $\varphi$ from $K$ onto $\mathbb{Z}\cup \{\infty\}$ satisfying:
(I) $\varphi(a)=\infty$ if and only if $a=0$
(II) $\varphi(ab)=\varphi(a)+\varphi(b)$
(III) $\varphi(a+b)\ge \min\{\varphi(a),\varphi(b)\}$
We want to show that $$R=\{z\in K\mid \varphi(z)\ge 0\}$$ is a DVR with maximal ideal $$\mathfrak{m}=\{z\in K\mid \varphi(z) > 0\}.$$ I proved that $R$ is a domain, that $z\in U(R)$ iff $\varphi(z)=0$ and that $\mathfrak{m}$ is an ideal of $R$. So $R$ is local with maximal ideal $\mathfrak{m}$. But I can't show that $R$ is not field. I tried to show that $\mathfrak{m}$ is not trivial, it doesn't help. By the equivalent conditions of DVRs I tried find an irreducible element $t$ of $R$ such that for every non zero element $z$ in $R$, exists $u\in U(R)$ and $n\in \mathbb{N}$: $z=ut^n$, but also in this cases must show that $\mathfrak{m}$ is not trivial for the existence of this $t$. Could you help me to show that $\mathfrak{m}$ is not trivial ideal of $R$?