I'm trying to prove the following three statements:
"Let $D$ be a euclidean domain with norm $N$, such that $N(a) \leq N(ab)$ for all nonzero $a,b \in D$.
(1) Prove that $N(a) \geq N(1)$ for all nonzero $a \in D$.
(2) Prove that an element $a \in D$ is a unit if and only if $N(a)=N(1)$.
(3) Suppose that $D=\mathbb{Z}[i]$. Prove that $N(a) \leq N(ab)$ for all nonzero $a,b \in \mathbb{Z}[i]$." In this case, $N(a)=|a|^2$.
Could anyone walk me through how to prove these?
\begin{align} N(1) & \le N(1b) = N(b) \\[6pt] \text{So } N(b) & \le N(b) \text{ for every } b\in D. \end{align} That takes care of question $1.$
$a\in D$ is a unit precisely if for some $b\in D$ you have $ab=1.$ So then you have \begin{align} N(1) & \le N(a) & & \text{by the result of question 1} \\[6pt] & \le N(ab) & & \text{(This was given at the outset.)} \\[6pt] & \le N(1) & & \text{since } ab=1. \end{align} That finishes off question $2.$