# Cyclotomic Fields

- [Introduction](introduction.md)

- [Creation Functions](creation.md)

  - [Creation of Cyclotomic Fields](creation.md#creation-of-cyclotomic-fields)

    - [`CyclotomicField(m): RngIntElt → FldCyc`](creation.md#function-cyclotomicfield-rngintelt)

    - [`CyclotomicPolynomial(m): RngIntElt → RngUPolElt`](creation.md#function-cyclotomicpolynomial-rngintelt)

    - [`MinimalField(a): FldRatElt → FldRat`](creation.md#function-minimalfield-fldratelt)

    - [`MinimalField(a): FldCycElt → FldCyc`](creation.md#function-minimalfield-fldcycelt)

    - [`MinimalField(a): RngCycElt → RngCyc`](creation.md#function-minimalfield-rngcycelt)

    - [`MinimalCyclotomicField(a): FldRatElt → FldRat`](creation.md#function-minimalcyclotomicfield-fldratelt)

    - [`MinimalCyclotomicField(a): FldCycElt → FldCyc`](creation.md#function-minimalcyclotomicfield-fldcycelt)

    - [`MinimalCyclotomicField(a): RngCycElt → RngCyc`](creation.md#function-minimalcyclotomicfield-rngcycelt)

    - [`MinimalField(S): [ FldCycElt ] → FldCyc`](creation.md#function-minimalfield-fldcycelt-2)

    - [`MinimalField(S): { FldCycElt } → FldCyc`](creation.md#function-minimalfield-fldcycelt-3)

    - [`MinimalField(S): [ RngCycElt ] → RngCyc`](creation.md#function-minimalfield-rngcycelt-2)

    - [`MinimalField(S): { RngCycElt } → RngCyc`](creation.md#function-minimalfield-rngcycelt-3)

    - [`MinimalCyclotomicField(S): [ FldCycElt ] → FldCyc`](creation.md#function-minimalcyclotomicfield-fldcycelt-2)

    - [`MinimalCyclotomicField(S): Setq → FldCyc`](creation.md#function-minimalcyclotomicfield-setq)

    - [`MinimalCyclotomicField(S): { FldCycElt } → FldCyc`](creation.md#function-minimalcyclotomicfield-fldcycelt-3)

    - [`MinimalCyclotomicField(S): [ RngCycElt ] → RngCyc`](creation.md#function-minimalcyclotomicfield-rngcycelt-2)

    - [`MinimalCyclotomicField(S): { RngCycElt } → RngCyc`](creation.md#function-minimalcyclotomicfield-rngcycelt-3)

    - [`Example: creation`](creation.md#example-ex-644033)

  - [Creation of Elements](creation.md#creation-of-elements)

    - [`RootOfUnity(n): RngIntElt → FldCycElt`](creation.md#function-rootofunity-rngintelt)

    - [`RootOfUnity(n, K): RngIntElt, FldCyc → FldCycElt`](creation.md#function-rootofunity-rngintelt-fldcyc)

    - [`Minimise(~a): FldCycElt`](creation.md#function-minimise-fldcycelt-ref)

    - [`Minimize(~a): FldCycElt`](creation.md#function-minimize-fldcycelt-ref)

    - [`Minimise(~a): RngCycElt`](creation.md#function-minimise-rngcycelt-ref)

    - [`Minimize(~a): RngCycElt`](creation.md#function-minimize-rngcycelt-ref)

    - [`Minimise(~s): [ FldCycElt ]`](creation.md#function-minimise-fldcycelt-ref-2)

    - [`Minimize(~s): [ FldCycElt ]`](creation.md#function-minimize-fldcycelt-ref-2)

    - [`Minimise(~s): [ RngCycElt ]`](creation.md#function-minimise-rngcycelt-ref-2)

    - [`Minimize(~s): [ RngCycElt ]`](creation.md#function-minimize-rngcycelt-ref-2)

    - [`Minimise(a): FldCycElt → RngElt`](creation.md#function-minimise-fldcycelt)

    - [`Minimize(a): FldCycElt → RngElt`](creation.md#function-minimize-fldcycelt)

    - [`Minimise(a): RngCycElt → RngElt`](creation.md#function-minimise-rngcycelt)

    - [`Minimize(a): RngCycElt → RngElt`](creation.md#function-minimize-rngcycelt)

    - [`Minimise(s): { FldCycElt } → { RngElt }`](creation.md#function-minimise-fldcycelt-2)

    - [`Minimize(s): { FldCycElt } → { RngElt }`](creation.md#function-minimize-fldcycelt-2)

    - [`Minimise(s): { RngCycElt } → { RngElt }`](creation.md#function-minimise-rngcycelt-2)

    - [`Minimize(s): { RngCycElt } → { RngElt }`](creation.md#function-minimize-rngcycelt-2)

- [Structure Operations](structure.md)

  - [Invariants](structure.md#invariants)

    - [`Conductor(K): FldCyc → RngIntElt, [RngIntElt]`](structure.md#function-conductor-fldcyc)

    - [`CyclotomicOrder(K): FldCyc → RngIntElt`](structure.md#function-cyclotomicorder-fldcyc)

    - [`CyclotomicOrder(K): FldRat → RngIntElt`](structure.md#function-cyclotomicorder-fldrat)

    - [`CyclotomicAutomorphismGroup(K): FldCyc → GrpAb, Map`](structure.md#function-cyclotomicautomorphismgroup-fldcyc)

    - [`CyclotomicRelativeField(k, K): FldCyc, FldCyc → FldNum`](structure.md#function-cyclotomicrelativefield-fldcyc-fldcyc)

- [Element Operations](element.md)

  - [Predicates on Elements](element.md#predicates-on-elements)

    - [`IsReal(a): FldCycElt → BoolElt`](element.md#function-isreal-fldcycelt)

    - [`IsReal(a): RngCycElt → BoolElt`](element.md#function-isreal-rngcycelt)

  - [Conjugates](element.md#conjugates)

    - [`ComplexConjugate(a): FldCycElt → FldCycElt`](element.md#function-complexconjugate-fldcycelt)

    - [`ComplexConjugate(a): RngCycElt → RngCycElt`](element.md#function-complexconjugate-rngcycelt)

    - [`Conjugate(a, n): FldCycElt, RngIntElt → FldCycElt`](element.md#function-conjugate-fldcycelt-rngintelt)

    - [`Conjugate(a, n): RngCycElt, RngIntElt → RngCycElt`](element.md#function-conjugate-rngcycelt-rngintelt)

    - [`Conjugate(a, r): FldCycElt, FldCycElt → FldCycElt`](element.md#function-conjugate-fldcycelt-fldcycelt)

    - [`Conjugate(a, r): RngCycElt, RngCycElt → RngCycElt`](element.md#function-conjugate-rngcycelt-rngcycelt)

    - [`Example: Gaussian Periods`](element.md#example-ex-591c1c)