$\color{black}{Background:}$
$\textbf{Definition:}$ An integral domain $R$ is a Euclidean domain if there is a function $\delta$ from the nonzero elements of $R$ to the nonnegative integers with these properties;
$(i)$ If $a$ and $b$ are nonzero elements of $R,$ then $\delta(a)\leq \delta(ab).$
$(ii)$ If $a,b\in R$ and $b\neq 0_R,$ then there exist $q,r\in R$ such that $a=bq+r$ and either $r=0_R$ or $\delta(r) < \delta(b).$
$\textbf{Exercise 14:}$ Let $R$ be a field. Prove that $R$ is a Euclidean domain with the function $\delta$ given by $\delta(a)=0$ for each nonzero $a\in R.$
$\textbf{Exercise 29:}$ Let $R$ be a Euclidean domain. If the function $\delta$ is a constant function, prove that $R$ is a field.
$\color{black}{Questions:}$
I have a question about exercise 14 above. I have seen in some proof of exercise 14 that the $\delta(a)=1$ instead of $\delta(a)=0$. I am just wondering if it matters. If $\delta(a)=1$, the proof doesn't require that I need to worry about $r\neq 0.$ Also, in light of exercise 29, it seems that it really doesn't matter.
Thank you in advance
None of the requirements on $δ$ (apart from having a codomain of natural numbers) say anything about what values it actually takes, only whether it's larger or smaller on some points than on others. If $x↦δ(x)$ fulfills all the requirements, then so does $x↦δ(x)+1$. And if $δ(x)$ is never equal to $0$, so does $x↦δ(x)−1$ (and so does any other manipulation you can think of that preserves the order and doesn't take us out of the codomain of natural numbers).