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The extension $\mathbb{Q}(\zeta_n)|\mathbb{Q}$ is abelian of group $(\mathbb{Z}/n\mathbb{Z})^\times$ so class field theory tells you everything about the prime ideals in $\mathbb{Z}[\zeta_n]$, the ring of integers of $\mathbb{Q}(\zeta_n)$.

You should try to do the cases $n=3,4$ by hand.

As for the group $\mathbb{Z}[\zeta_n]^\times$, an explicit subgroup of "cyclotomic units" can be constructed which has finite index.

Any book on Cyclotomic Fields (Lang, Washington) should help. For a start, you can look up Chapter VI of Fröhlich-Taylor.
show/hide this revision's text 2 Added the advice about n=3,4.

The extension $\mathbb{Q}(\zeta_n)|\mathbb{Q}$ is abelian of group $(\mathbb{Z}/n\mathbb{Z})^\times$ so class field theory tells you everything about the prime ideals in $\mathbb{Z}[\zeta_n]$, the ring of integers of $\mathbb{Q}(\zeta_n)$.

You should try to do the cases $n=3,4$ by hand.

As for the group $\mathbb{Z}[\zeta_n]^\times$, an explicit subgroup of "cyclotomic units" can be constructed which has finite index. Any book on Cyclotomic Fields (Lang, Washington) should help.
show/hide this revision's text 1

The extension $\mathbb{Q}(\zeta_n)|\mathbb{Q}$ is abelian of group $(\mathbb{Z}/n\mathbb{Z})^\times$ so class field theory tells you everything about the prime ideals in $\mathbb{Z}[\zeta_n]$, the ring of integers of $\mathbb{Q}(\zeta_n)$.

As for the group $\mathbb{Z}[\zeta_n]^\times$, an explicit subgroup of "cyclotomic units" can be constructed which has finite index. Any book on Cyclotomic Fields (Lang, Washington) should help.