# Structure Operations

In cyclotomic fields the generic ring functions are supported (see Chapter [Introduction to Rings](../../BasicRings/IntroductionToRings/index-introduction-to-rings.md#rngintro)). The functions listed below are those functions for cyclotomic fields which are additional to those for number fields. For the list of functions applying to general number fields see Section [Creation Functions](../NumberFields/creation.md#fldnum-create) and Section [Structure Operations](../NumberFields/operation.md#fldnum-str-op).

## Invariants

### `Conductor(K): FldCyc -> RngIntElt, [RngIntElt]`

The smallest $n$ such that the field $K$ is contained in ${\mathbb{Q}}(\zeta_n)$; for a cyclotomic field that is either the ‘cyclotomic order’ $m$ (see below) or half that, depending on whether $m \equiv 2 \bmod 4$. The second return value is a sequence of the ramified real places of $K$.

### `CyclotomicOrder(K): FldCyc -> RngIntElt`

### `CyclotomicOrder(K): FldRat -> RngIntElt`

The value of $m$ for the cyclotomic field ${\mathbb{Q}}(\zeta_m)$. Note that this will be the $m$ with which the cyclotomic field was created.

### `CyclotomicAutomorphismGroup(K): FldCyc -> GrpAb, Map`

Returns the automorphism group of $K$ as an abstract abelian group $G$ and a map from $G$ into the set of all automorphisms. Note that similar functionality is also available through [`AutomorphismGroup`](../GaloisGroupsAndAutomorphisms/automorphism.md#function-rngord-automorphismgroup) however, this function returns an abelian group and uses the fact that the automorphism group is already determined by the conductor.

### `CyclotomicRelativeField(k, K): FldCyc, FldCyc -> FldNum`

Given two cyclotomic fields $k\subseteq K$ a number field $L/k$ is computed that is isomorphic to $K$.