57.2 Operations for Abelian Number Fields

For operations for cyclotomics, e.g., Conductor (see Conductor), see Chapter Cyclotomic Numbers.

  • IsNumberField( F ) P

    A number field is a finite extension of a prime field in characteristic zero.

  • IsAbelianNumberField( F ) P

    An abelian number field is a number field that is a Galois extension of the prime field with abelian Galois group (see GaloisGroup.field).

  • GaloisStabilizer( F ) A

    For an abelian number field F, GaloisStabilizer returns the set of all integers k in the range from 1 to the conductor of F such that the field automorphism induced by raising roots of unity in F to the k-th power acts trivially on F.

  • ComplexConjugate( z ) A

    For a cyclotomic number z, ComplexConjugate returns GaloisCyc( z, -1 ). For a quaternion z = c1 e + c2 i + c3 j + c4 k, ComplexConjugate returns c1 e - c2 i - c3 j - c4 k.

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    GAP 4 manual
    February 2000