The wikipedia page on unique factorization domains gives the following example.
What does the notation Z[something] mean?
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Given a ring extension $S\supseteq R$ and an element $\alpha\in S$ one denotes by $$ R[\alpha] $$ the smallest subring of $S$ containg $R$ and $\alpha$.
In the above example, $R=\Bbb Z$ and $\alpha=\zeta_n$, a primitive $n$-th root of unity.
The ring $\Bbb Z[\zeta_n]$ is the ring of integers in the cyclotomic field $\Bbb Q(\zeta_n)$.
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