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.

References

• A. Fröhlich; M.J. Taylor (1991). Algebraic number theory. Cambridge University Press. ISBN 0-521-36664-X. 
• I.N. Stewart; D.O. Tall (1979). Algebraic number theory. Chapman and Hall. ISBN 0-412-13840-9. 
• Pierre Samuel (1972). Algebraic number theory. Hermann/Kershaw.