# what is Z[x] in the sense of a unique factorization domain

What does the notation Z[something] mean?

• 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? Mar 10 '20 at 10:25
• 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. Mar 10 '20 at 10:28

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 $$\Bbb Q(\zeta_n)$$.