Let $D$ with $\phi: D-\{0\}\to\mathbb{N}$ be an Euclidean domain. Suppose $\phi (a+b)\leq max${$\phi (a),\phi (b)$} for all $a,b\in D$. Prove that $D$ is either a field or isomorphic to a polynomial ring over a field.
Here is the full proof link.There's a part in the proof I don't understand.
Why the author can assume $g(1)=0$? The explanation behind doesn't make sense to me. What is $a$ exactly here? Could someone help me?
As said in the article you linked, let $g'(a)=g(a)-g(1)$, this gives you a map $g':D\setminus\{0\}\longrightarrow\mathbb{N}$ such that $g'(1)=0$ and satisfying the same properties as $g$, namely $$ \forall a,b\in D\setminus\{0\},g'(ab)\geqslant g'(a)\geqslant 0 $$ $$ \forall a\in D\setminus\{0\},\forall b\in D,\exists q,r\in D,b=qa+r, \text{ with } r=0 \text{ or } g'(r)<g'(a). $$ This immediately follows from the fact that $g$ satisfies these properties. Now, you can replace $g$ by $g'$ so that $g(1)=0$.