= Solution
Every <Euclidean domain> is a <principal ideal domain>: in a nonzero ideal choose an element of least positive Euclidean value and divide every other element by it; minimality forces every remainder to vanish. Every principal ideal domain is a <unique factorization domain>: the ascending-chain condition supplies factorizations into irreducibles, and <Euclid lemma> follows because every irreducible generates a prime ideal. Therefore
$$
\boxed{\mathbb Z[\omega]\text{ is a unique factorization domain}}.
$$
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