# Cyclotomic field  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] This editable Main Article is under development and subject to a disclaimer. [edit intro]

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 factorisation of the polynomial $X^{n}-1$ modulo p, which in turn depends on the highest common factor of p-1 and n.

## Galois group

The minimal polynomial for ζ is the n-th cyclotomic polynomial $\Phi _{n}(X)$ , which is a factor of $X^{n}-1$ . Since the powers of ζ are the roots of the latter polynomial, F is a splitting field for $\Phi _{n}(X)$ and hence a Galois extension. The Galois group is isomorphic to the multiplicative group, $(\mathbf {Z} /n\mathbf {Z} )^{*}$ via

$a{\bmod {n}}\leftrightarrow \sigma _{a}=(\zeta \mapsto \zeta ^{a}).\,$ 