= Solution
Let $R$ be a <Euclidean domain> and $I$ a nonzero <ideal>. Choose $d\in I\setminus\{0\}$ minimizing the Euclidean size. For any $a\in I$, <Euclidean division> gives $a=qd+r$, where either $r=0$ or $r$ has smaller size than $d$. But $r=a-qd\in I$, so minimality forces $r=0$. Consequently $I=(d)$. The zero <ideal> is $(0)$, proving that \b[every <Euclidean domain> is a <principal ideal domain>].
A commutative ring is <Noetherian> if every <ideal> is finitely generated; equivalently, every ascending chain of <ideals> stabilizes. To see the equivalence, the union of an ascending chain is an <ideal>. Finitely many generators for that union already belong to one member of the chain, after which the chain is constant. Conversely, an <ideal> which is not finitely generated permits successive choices outside the <ideals> generated by previous choices, producing a strictly ascending chain.
Every <principal ideal domain> is therefore <Noetherian>. The given <Euclidean domain> property of the <Gaussian integers> yields
$$
\boxed{\mathbb Z[i]\text{ is Noetherian}.}
$$
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