Cyclotomic field

In mathematics, a cyclotomic field is a field which is an extension generated by roots of unity. If ζ denotes an n-th root of unity, then the n-th cyclotomic field F is the field extension $$\mathbf{Q}(\zeta)$$.

Ring of integers
As above, we take ζ to denote an n-th root of unity. The maximal order of F is
 * $$O_F = \mathbf{Z}[\zeta]. \,$$

Splitting of primes
A prime p ramifies iff p divides n. Otherwise, the splitting of p depends on the common factor of p-1 and n.