Cyclotomic Fields

• Introduction
• Creation Functions
  • Creation of Cyclotomic Fields
    • CyclotomicField(m): RngIntElt → FldCyc
    • CyclotomicPolynomial(m): RngIntElt → RngUPolElt
    • MinimalField(a): FldRatElt → FldRat
    • MinimalField(a): FldCycElt → FldCyc
    • MinimalField(a): RngCycElt → RngCyc
    • MinimalCyclotomicField(a): FldRatElt → FldRat
    • MinimalCyclotomicField(a): FldCycElt → FldCyc
    • MinimalCyclotomicField(a): RngCycElt → RngCyc
    • MinimalField(S): [ FldCycElt ] → FldCyc
    • MinimalField(S): { FldCycElt } → FldCyc
    • MinimalField(S): [ RngCycElt ] → RngCyc
    • MinimalField(S): { RngCycElt } → RngCyc
    • MinimalCyclotomicField(S): [ FldCycElt ] → FldCyc
    • MinimalCyclotomicField(S): Setq → FldCyc
    • MinimalCyclotomicField(S): { FldCycElt } → FldCyc
    • MinimalCyclotomicField(S): [ RngCycElt ] → RngCyc
    • MinimalCyclotomicField(S): { RngCycElt } → RngCyc
    • Example: creation
  • Creation of Elements
    • RootOfUnity(n): RngIntElt → FldCycElt
    • RootOfUnity(n, K): RngIntElt, FldCyc → FldCycElt
    • Minimise(~a): FldCycElt
    • Minimize(~a): FldCycElt
    • Minimise(~a): RngCycElt
    • Minimize(~a): RngCycElt
    • Minimise(~s): [ FldCycElt ]
    • Minimize(~s): [ FldCycElt ]
    • Minimise(~s): [ RngCycElt ]
    • Minimize(~s): [ RngCycElt ]
    • Minimise(a): FldCycElt → RngElt
    • Minimize(a): FldCycElt → RngElt
    • Minimise(a): RngCycElt → RngElt
    • Minimize(a): RngCycElt → RngElt
    • Minimise(s): { FldCycElt } → { RngElt }
    • Minimize(s): { FldCycElt } → { RngElt }
    • Minimise(s): { RngCycElt } → { RngElt }
    • Minimize(s): { RngCycElt } → { RngElt }
• Structure Operations
  • Invariants
    • Conductor(K): FldCyc → RngIntElt, [RngIntElt]
    • CyclotomicOrder(K): FldCyc → RngIntElt
    • CyclotomicOrder(K): FldRat → RngIntElt
    • CyclotomicAutomorphismGroup(K): FldCyc → GrpAb, Map
    • CyclotomicRelativeField(k, K): FldCyc, FldCyc → FldNum
• Element Operations
  • Predicates on Elements
    • IsReal(a): FldCycElt → BoolElt
    • IsReal(a): RngCycElt → BoolElt
  • Conjugates
    • ComplexConjugate(a): FldCycElt → FldCycElt
    • ComplexConjugate(a): RngCycElt → RngCycElt
    • Conjugate(a, n): FldCycElt, RngIntElt → FldCycElt
    • Conjugate(a, n): RngCycElt, RngIntElt → RngCycElt
    • Conjugate(a, r): FldCycElt, FldCycElt → FldCycElt
    • Conjugate(a, r): RngCycElt, RngCycElt → RngCycElt
    • Example: Gaussian Periods