# Cyclotomic field

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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 ${\displaystyle \mathbf {Q} (\zeta )}$.

## Ring of integers

As above, we take ζ to denote an n-th root of unity. The maximal order of F is

${\displaystyle O_{F}=\mathbf {Z} [\zeta ].\,}$

## Splitting of primes

A prime p ramifies iff p divides n. Otherwise, the splitting of p depends on the factorisation of the polynomial ${\displaystyle X^{n}-1}$ 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 ${\displaystyle \Phi _{n}(X)}$, which is a factor of ${\displaystyle X^{n}-1}$. Since the powers of ζ are the roots of the latter polynomial, F is a splitting field for ${\displaystyle \Phi _{n}(X)}$ and hence a Galois extension. The Galois group is isomorphic to the multiplicative group, ${\displaystyle (\mathbf {Z} /n\mathbf {Z} )^{*}}$ via

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