[ref] 58 Abelian Number Fields

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# 58 Abelian Number Fields

### Sections

1. Integral Bases for Abelian Number Fields
2. Operations for Abelian Number Fields
3. Galois Groups of Abelian Number Fields
4. Gaussians

- `CyclotomicField(` n`) F`
- `CyclotomicField(` gens`) F`
- `CyclotomicField(` subfield`,` n`) F`
- `CyclotomicField(` subfield`,` gens`) F`

  The first version creates the n-th cyclotomic field. The second version creates the cyclotomic field generated by gens. In both cases the field can be generated as an extension of a designated subfield.

- `AbelianNumberField(` n`,` stab`) F`

  fixed field of the group generated by stab (prime residues modulo n) in the cyclotomic field with conductor n.

- `GaussianRationals V`

  is the field Q(i) of Gaussian rationals.

- `IsGaussianRationals(` obj`) C`

  ## 58.1 Integral Bases for Abelian Number Fields

- `ZumbroichBase(` n`,` m`) F`

  is the set of exponents e for which `E(`n`)^`e`` belongs to the (generalized) Zumbroich base of the cyclotomic field Qn, viewed as vector space over Qm.

  The base, the base conversion and the reduction to the minimal cyclotomic field are described in Zum89.

  Note: For n º 2 mod 4 we have `ZumbroichBase(`n`, 1) = 2 * ZumbroichBase(`n`/2, 1)`. However, `List( ZumbroichBase(`n`, 1), x -> E(`n`)^x ) = List( ZumbroichBase(`n`/2, 1), x -> E(`n`/2)^x )`.

- `LenstraBase(` n`,` stabilizer`,` super`,` m`) F`

  is a list of lists of integers, each list indexing the exponents of an orbit of a subgroup of stabilizer on n-th roots of unity.

  super is a list representing a supergroup of stabilizer which shall act consistently with the action of stabilizer, i.e., each orbit of supergroup is a union of orbits of stabilizer.

  m is a positive integer. The basis described by the returned list is an integral basis over the cyclotomic field Qm.

  Note: The elements are in general not sets, since the first element is always an element of `ZumbroichBase(` n`,` m`)`; this property is used by `NF` and `Coefficients`.

  Note: stabilizer must not contain the stabilizer of a proper cyclotomic subfield of the n-th cyclotomic field.

  For details about the bases see Bre97.

  ## 58.2 Operations for Abelian Number Fields

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

- `IsNumberField(` F`) P`

  returns `true` if the field F is a number field, or `false` otherwise.

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

- `IsAbelianNumberField(` F`) P`

  returns the abelian number field of the field F.

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

  ## 58.3 Galois Groups of Abelian Number Fields

  The Galois automorphisms of the cyclotomic field Q(en ) (see Chapter Cyclotomic Numbers) are given by linear extension of the maps *k : en ® en k where 1 £ k < n and `Gcd(` n`,` k`) = 1` hold (see GaloisCyc). Note that this action is not equal to exponentiation of cyclotomics, i.e., for general cyclotomics z, z*k is different from zk:

      gap> ( E(5) + E(5)^4 )^2; GaloisCyc( E(5) + E(5)^4, 2 );
      -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4
      E(5)^2+E(5)^3

  For `Gcd`( n , k ) ¹ 1, the map en ® en k is not a field automorphism but only a linear map.

      gap> GaloisCyc( E(5)+E(5)^4, 5 ); GaloisCyc( ( E(5)+E(5)^4 )^2, 5 );
      2
      -6

- `ANFAutomorphism(` F`,` k`) F`

  Let F be an abelian number field F and k an integer. If k is coprime to the conductor (see Conductor) of F then `ANFAutomorphism` returns the automorphism of F defined as the linear extension of the map that raises each root of unity in F to its k-th power, otherwise an error is signalled.

      gap> f:= CF(25);
      CF(25)
      gap> alpha:= ANFAutomorphism( f, 2 );
      ANFAutomorphism( CF(25), 2 )
      gap> alpha^2;
      ANFAutomorphism( CF(25), 4 )
      gap> Order( alpha );
      20
      gap> E(5)^alpha;
      E(5)^2

  The Galois group Gal( Q(en ), Q) of the field extension Q(en ) / Q is isomorphic to the group (Z/ n Z)* of prime residues modulo n, via the isomorphism from (Z/ n Z)* to Gal( Q(en ), Q) defined by k + n Z® ( z ® z*k ).

  The Galois group of the field extension Q(en ) / L with any abelian number field L Í Q(en ) is simply the factor group of Gal( Qn , Q) modulo the stabilizer of L, and the Galois group of L / L¢, with L¢ an abelian number field contained in L is the subgroup in this group that stabilizes L¢. These groups are easily described in terms of (Z/ n Z)*. Generators of (Z/ n Z)* can be computed using `GeneratorsPrimeResidues` (see GeneratorsPrimeResidues).

      gap> f:= CF(15);
      CF(15)
      gap> g:= GaloisGroup( f );
      <group with 2 generators>
      gap> Size( g ); IsCyclic( g ); IsAbelian( g );
      8
      false
      true
      gap> Action( g, NormalBase( f ), OnPoints );
      Group([ (1,6)(2,4)(3,8)(5,7), (1,4,3,7)(2,8,5,6) ])
      gap> GeneratorsOfGroup( g );
      [ ANFAutomorphism( CF(15), 11 ), ANFAutomorphism( CF(15), 7 ) ]

  ## 58.4 Gaussians

- `GaussianIntegers V`

  is the ring of Gaussian integers. This is the subring Z[i] of the complex numbers, where i is a square root of -1.

- `IsGaussianIntegers(` obj`) C`

  is the defining category for the domain `GaussianIntegers`.

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  GAP 4 manual
  May 2002