Euclidean domains with multiplicative and super triangular norms 
I want to prove that if the norm function $N$ of a Euclidean domain $R$ satisfies the conditions

*

*$N(ab)=N(a)N(b)$


*$N(a+b) \le max\{N(a),N(b)\}$
then $R$ is a field or R is a homomorphic image of a polynomial ring $F[x]$ where $F$ is any field.

I know the converse is true, for if $R$ is a field then the trivial norm satisfies the conditions; and if $R=F[x]$ then we can define $N(f)=2^{\deg(f)}$.
I can prove that condition (1) implies that a is a unit iff $N(a)=1$.
but I have no good idea for how to use condition (2).
can anybody help me?

This is a sketch of the proof. Nice exercise from {Algebra , N.Jacobson , V.1 , p149}.
From (1) we can deduce that $a$ is invertible iff $N(a)=1$.
From (2) we can deduce that the sum of two invertible elements is again invertible or $0$. Hence, the set of invertible elements together with $0$ is a field we call $F$.
if $R=F$ we are done. Otherwise, let $t \in R$ be a non-invertible element with the least possible norm. Using (1) we can get a base-$t$ representation for any element of $R$ with coefficients from $F$, so we found a surjective homomorphism between $R$ and $F[t]$.
