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The wikipedia page on unique factorization domains gives the following example. enter image description here

What does the notation Z[something] mean?

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  • $\begingroup$ is Z[x] the same as the set of all ax+b where a and b come from Z ? or is it more subtle than that? $\endgroup$
    – Jim Newton
    Mar 10 '20 at 10:25
  • $\begingroup$ No, it is more complicated. Think of fields first, i.e., of $\Bbb Q(\zeta_n)$. Since this is a vector space of dimension $\phi(n)$ over $\Bbb Q$, we have a basis $(1,\zeta_n,\zeta_n^2,\cdots)$, which can have more than $2$ elements. $\endgroup$ Mar 10 '20 at 10:28
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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 $\Bbb Q(\zeta_n)$.

References form this site:

Adjoining an element to a ring

Adjoining to a ring

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