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# 58 Abelian Number Fields

### Sections

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

An abelian number field is a field in characteristic zero that is a finite dimensional normal extension of its prime field such that the Galois group is abelian. In GAP, one implementation of abelian number fields is given by fields of cyclotomic numbers (see Chapter Cyclotomic Numbers). Note that abelian number fields can also be constructed with the more general `AlgebraicExtension` (see AlgebraicExtension), a discussion of advantages and disadvantages can be found in Internally Represented Cyclotomics. The functions described in this chapter have been developed for fields whose elements are in the filter `IsCyclotomic` (see IsCyclotomic), they may or may not work well for abelian number fields consisting of other kinds of elements.

Throughout this chapter, Qn will denote the cyclotomic field generated by the field Q of rationals together with n-th roots of unity.

In Construction of Abelian Number Fields, constructors for abelian number fields are described, Operations for Abelian Number Fields introduces operations for abelian number fields, Integral Bases of Abelian Number Fields deals with the vector space structure of abelian number fields, and Galois Groups of Abelian Number Fields describes field automorphisms of abelian number fields,

## 58.1 Construction of Abelian Number Fields

Besides the usual construction using `Field` or `DefaultField` (see Operations for Abelian Number Fields), abelian number fields consisting of cyclotomics can be created with `CyclotomicField` and `AbelianNumberField`.

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

  The first version creates the n-th cyclotomic field Qn. The second version creates the smallest cyclotomic field containing the elements in the list gens. In both cases the field can be generated as an extension of a designated subfield subfield (cf. Integral Bases of Abelian Number Fields).

  `CyclotomicField` can be abbreviated to `CF`, this form is used also when GAP prints cyclotomic fields.

  Fields constructed with the one argument version of `CF` are stored in the global list `CYCLOTOMIC_FIELDS`, so repeated calls of `CF` just fetch these field objects after they have been created once.

      gap> CyclotomicField( 5 );  CyclotomicField( [ Sqrt(3) ] );
      CF(5)
      CF(12)
      gap> CF( CF(3), 12 );  CF( CF(4), [ Sqrt(7) ] );
      AsField( CF(3), CF(12) )
      AsField( GaussianRationals, CF(28) )

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

  For a positive integer n and a list stab of prime residues modulo n, `AbelianNumberField` returns the fixed field of the group described by stab (cf. GaloisStabilizer), in the n-th cyclotomic field. `AbelianNumberField` is mainly thought for internal use and for printing fields in a standard way; `Field` (see Field, cf. also Operations for Abelian Number Fields) is probably more suitable if one knows generators of the field in question.

  `AbelianNumberField` can be abbreviated to `NF`, this form is used also when GAP prints abelian number fields.

  Fields constructed with `NF` are stored in the global list `ABELIAN_NUMBER_FIELDS`, so repeated calls of `NF` just fetch these field objects after they have been created once.

      gap> NF( 7, [ 1 ] );
      CF(7)
      gap> f:= NF( 7, [ 1, 2 ] );  Sqrt(-7); Sqrt(-7) in f;
      NF(7,[ 1, 2, 4 ])
      E(7)+E(7)^2-E(7)^3+E(7)^4-E(7)^5-E(7)^6
      true

- `GaussianRationals V`
- `IsGaussianRationals(` obj`) C`

  `GaussianRationals` is the field Q4 = Q(√{−1}) of Gaussian rationals, as a set of cyclotomic numbers, see Chapter Cyclotomic Numbers for basic operations. This field can also be obtained as `CF(4)` (see CyclotomicField).

  The filter `IsGaussianRationals` returns `true` for the GAP object `GaussianRationals`, and `false` for all other GAP objects.

  (For details about the field of rationals, see Chapter Rationals.)

      gap> CF(4) = GaussianRationals;
      true
      gap> Sqrt(-1) in GaussianRationals;
      true

  ## 58.2 Operations for Abelian Number Fields

  For operations for elements of abelian number fields, e.g., `Conductor` (see Conductor) or `ComplexConjugate` (see ComplexConjugate), see Chapter Cyclotomic Numbers.

  For a dense list l of cyclotomics, `DefaultField` (see DefaultField) returns the smallest cyclotomic field containing all entries of l, `Field` (see Field) returns the smallest field containing all entries of l, which need not be a cyclotomic field. In both cases, the fields represent vector spaces over the rationals (see Integral Bases of Abelian Number Fields).

      gap> DefaultField( [ E(5) ] ); DefaultField( [ E(3), ER(6) ] );
      CF(5)
      CF(24)
      gap> Field( [ E(5) ] ); Field( [ E(3), ER(6) ] );
      CF(5)
      NF(24,[ 1, 19 ])

  Factoring of polynomials over abelian number fields consisting of cyclotomics works in principle but is not very efficient if the degree of the field extension is large.

      gap> x:= Indeterminate( CF(5) );
      x_1
      gap> Factors( PolynomialRing( Rationals ), x^5-1 );
      [ x_1-1, x_1^4+x_1^3+x_1^2+x_1+1 ]
      gap> Factors( PolynomialRing( CF(5) ), x^5-1 );
      [ x_1-1, x_1+(-E(5)), x_1+(-E(5)^2), x_1+(-E(5)^3), x_1+(-E(5)^4) ]

- `IsNumberField(` F`) P`

  returns `true` if the field F is a finite dimensional extension of a prime field in characteristic zero, and `false` otherwise.

- `IsAbelianNumberField(` F`) P`

  returns `true` if the field F is a number field (see IsNumberField) that is a Galois extension of the prime field, with abelian Galois group (see GaloisGroup!of field).

- `IsCyclotomicField(` F`) P`

  returns `true` if the field F is a cyclotomic field, i.e., an abelian number field (see IsAbelianNumberField) that can be generated by roots of unity.

      gap> IsNumberField( CF(9) ); IsAbelianNumberField( Field( [ ER(3) ] ) );
      true
      true
      gap> IsNumberField( GF(2) );
      false
      gap> IsCyclotomicField( CF(9) );
      true
      gap> IsCyclotomicField( Field( [ Sqrt(-3) ] ) );
      true
      gap> IsCyclotomicField( Field( [ Sqrt(3) ] ) );
      false

- `GaloisStabilizer(` F`) A`

  Let F be an abelian number field (see IsAbelianNumberField) with conductor n, say. (This means that the n-th cyclotomic field is the smallest cyclotomic field containing F, see Conductor.) `GaloisStabilizer` returns the set of all those integers k in the range from 1 to n such that the field automorphism induced by raising n-th roots of unity to the k-th power acts trivially on F.

      gap> r5:= Sqrt(5);
      E(5)-E(5)^2-E(5)^3+E(5)^4
      gap> GaloisCyc( r5, 4 ) = r5;  GaloisCyc( r5, 2 ) = r5;
      true
      false
      gap> GaloisStabilizer( Field( [ r5 ] ) );
      [ 1, 4 ]

  ## 58.3 Integral Bases of Abelian Number Fields

  Each abelian number field is naturally a vector space over Q. Moreover, if the abelian number field F contains the n-th cyclotomic field Qn then F is a vector space over Qn. In GAP, each field object represents a vector space object over a certain subfield S, which depends on the way F was constructed. The subfield S can be accessed as the value of the attribute `LeftActingDomain` (see LeftActingDomain).

  The return values of `NF` (see AbelianNumberField) and of the one argument versions of `CF` (see CyclotomicField) represent vector spaces over Q, and the return values of the two argument version of `CF` represent vector spaces over the field that is given as the first argument. For an abelian number field F and a subfield S of F, a GAP object representing F as a vector space over S can be constructed using `AsField` (see AsField).

  Let F be the cyclotomic field Qn, represented as a vector space over the subfield S. If S is the cyclotomic field Qm, with m a divisor of n, then `CanonicalBasis(` F`)` returns the Zumbroich basis of F relative to S, which consists of the roots of unity `E``(`n `)`i where i is an element of the list `ZumbroichBase(` n`,` m`)` (see ZumbroichBase). If S is an abelian number field that is not a cyclotomic field then `CanonicalBasis(` F`)` returns a normal S-basis of F, i.e., a basis that is closed under the field automorphisms of F.

  Let F be the abelian number field `NF(` n`,` stab`)`, with conductor n, that is itself not a cyclotomic field, represented as a vector space over the subfield S. If S is the cyclotomic field Qm, with m a divisor of n, then `CanonicalBasis(` F`)` returns the Lenstra basis of F relative to S that consists of the sums of roots of unity described by `LenstraBase(` n`,` stab`,` stab`,` m`)` (see LenstraBase). If S is an abelian number field that is not a cyclotomic field then `CanonicalBasis(` F`)` returns a normal S-basis of F.

      gap> f:= CF(8);;   # a cycl. field over the rationals
      gap> b:= CanonicalBasis( f );;  BasisVectors( b );
      [ 1, E(8), E(4), E(8)^3 ]
      gap> Coefficients( b, Sqrt(-2) );
      [ 0, 1, 0, 1 ]
      gap> f:= AsField( CF(4), CF(8) );;  # a cycl. field over a cycl. field
      gap> b:= CanonicalBasis( f );;  BasisVectors( b );
      [ 1, E(8) ]
      gap> Coefficients( b, Sqrt(-2) );
      [ 0, 1+E(4) ]
      gap> f:= AsField( Field( [ Sqrt(-2) ] ), CF(8) );;
      gap> # a cycl. field over a non-cycl. field
      gap> b:= CanonicalBasis( f );;  BasisVectors( b );
      [ 1/2+1/2*E(8)-1/2*E(8)^2-1/2*E(8)^3, 1/2-1/2*E(8)+1/2*E(8)^2+1/2*E(8)^3 ]
      gap> Coefficients( b, Sqrt(-2) );
      [ E(8)+E(8)^3, E(8)+E(8)^3 ]
      gap> f:= Field( [ Sqrt(-2) ] );   # a non-cycl. field over the rationals
      NF(8,[ 1, 3 ])
      gap> b:= CanonicalBasis( f );;  BasisVectors( b );
      [ 1, E(8)+E(8)^3 ]
      gap> Coefficients( b, Sqrt(-2) );
      [ 0, 1 ]

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

  Let n and m be positive integers, such that m divides n. `ZumbroichBase` returns the set of exponents i for which `E(`n`)^`i`` belongs to the (generalized) Zumbroich basis of the cyclotomic field Qn, viewed as a vector space over Qm.

  This basis is defined as follows. Let P denote the set of prime divisors of n, n = ∏p ∈ P pνp, and m = ∏p ∈ P pμp with μp ≤ νp. Let en = `E``(`n `)`, and { en1j}j ∈ J ⊗{ en2k}k ∈ K = { en1j ·en2k}j ∈ J, k ∈ K.

  Then the basis is

  Bn,m = | ⊗ p ∈ P |  | νp−1 ⊗ k=μp | { epk+1j}j ∈ Jk,p where Jk,p = |          |
  { 0 }
  ;
  k=0, p=2
  { 0, 1 }
  ;
  k > 0, p=2
  { 1, …, p−1 }
  ;
  k = 0, p ≠ 2
  { − | p−1 2 | , …, | p−1 2 | }
  ;
  k > 0, p ≠ 2

  Bn,1 is equal to the basis of Qn over the rationals which is introduced in Zum89. Also the conversion of arbitrary sums of roots of unity into its basis representation, and the reduction to the minimal cyclotomic field are described in this thesis. (Note that the notation here is slightly different from that there.)

  Bn,m consists of roots of unity, it is an integral basis (that is, exactly the integral elements in Qn have integral coefficients w.r.t. Bn,m, cf. IsIntegralCyclotomic), it is a normal basis for squarefree n and closed under complex conjugation for odd n.

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

      gap> ZumbroichBase( 15, 1 ); ZumbroichBase( 12, 3 );
      [ 1, 2, 4, 7, 8, 11, 13, 14 ]
      [ 0, 3 ]
      gap> ZumbroichBase( 10, 2 ); ZumbroichBase( 32, 4 );
      [ 2, 4, 6, 8 ]
      [ 0, 1, 2, 3, 4, 5, 6, 7 ]

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

  Let n and m be positive integers, such that m divides n, stabilizer be a list of prime residues modulo n, which describes a subfield of the n-th cyclotomic field (see GaloisStabilizer), and super be a list representing a supergroup of the group given by stabilizer.

  `LenstraBase` returns a list [ b1, b2, …, bk ] of lists, each bi consisting of integers such that the elements ∑j ∈ bi `E``(``n``)`j form a basis of the abelian number field `NF(` n`,` stabilizer`)`, as a vector space over the m-th cyclotomic field (see AbelianNumberField).

  This basis is an integral basis, that is, exactly the integral elements in `NF(` n`,` stabilizer`)` have integral coefficients. (For details about this basis, see Bre97.)

  If possible then the result is chosen such that the group described by super acts on it, consistently with the action of stabilizer, i.e., each orbit of super is a union of orbits of stabilizer. (A usual case is ``super`=` stabilizer``, so there is no additional condition.

  Note: The bi are in general not sets, since for ``stabilizer`=` super``, the first entry is always an element of `ZumbroichBase(` n`,` m`)`; this property is used by `NF` (see AbelianNumberField) and `Coefficients` (see Integral Bases of Abelian Number Fields).

  stabilizer must not contain the stabilizer of a proper cyclotomic subfield of the n-th cyclotomic field, i.e., the result must describe a basis for a field with conductor n.

      gap> LenstraBase( 24, [ 1, 19 ], [ 1, 19 ], 1 );
      [ [ 1, 19 ], [ 8 ], [ 11, 17 ], [ 16 ] ]
      gap> LenstraBase( 24, [ 1, 19 ], [ 1, 5, 19, 23 ], 1 );
      [ [ 1, 19 ], [ 5, 23 ], [ 8 ], [ 16 ] ]
      gap> LenstraBase( 15, [ 1, 4 ], PrimeResidues( 15 ), 1 );
      [ [ 1, 4 ], [ 2, 8 ], [ 7, 13 ], [ 11, 14 ] ]

  The first two results describe two bases of the field Q3(√6), the third result describes a normal basis of Q3(√5).

  ## 58.4 Galois Groups of Abelian Number Fields

  The field automorphisms of the cyclotomic field Qn (see Chapter Cyclotomic Numbers) are given by the linear maps ∗k on Qn that are defined by `E`(n)∗k = `E`(n)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.

  (In GAP, the image of a cyclotomic z under ∗k can be computed as `GaloisCyc`( z, k ).)

      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, the map `E`(n) → `E`(n)k does not define a field automorphism of Qn but only a Q-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 and k an integer that is coprime to the conductor (see Conductor) of F. Then `ANFAutomorphism` returns the automorphism of F that is defined as the linear extension of the map that raises each root of unity in F to its k-th power.

      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( Qn, Q) of the field extension Qn / Q is isomorphic to the group (Z/ n Z)∗ of prime residues modulo n, via the isomorphism (Z/ n Z)∗ → Gal( Qn, Q) that is defined by k + n Z→ ( z → z∗k ).

  The Galois group of the field extension Qn / L with any abelian number field L ⊆ Qn 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).

  In GAP, a field extension L / L′ is given by the field object L with `LeftActingDomain` value L′ (see Integral Bases of Abelian Number Fields).

      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) ])

  The following example shows Galois groups of a cyclotomic field and of a proper subfield that is not a cyclotomic field.

      gap> gens1:= GeneratorsOfGroup( GaloisGroup( CF(5) ) );
      [ ANFAutomorphism( CF(5), 2 ) ]
      gap> gens2:= GeneratorsOfGroup( GaloisGroup( Field( Sqrt(5) ) ) );
      [ ANFAutomorphism( NF(5,[ 1, 4 ]), 2 ) ]
      gap> Order( gens1[1] );  Order( gens2[1] );
      4
      2
      gap> Sqrt(5)^gens1[1] = Sqrt(5)^gens2[1];
      true

  The following example shows the Galois group of a cyclotomic field over a non-cyclotomic field.

      gap> g:= GaloisGroup( AsField( Field( [ Sqrt(5) ] ), CF(5) ) );
      <group with 1 generators>
      gap> gens:= GeneratorsOfGroup( g );
      [ ANFAutomorphism( AsField( NF(5,[ 1, 4 ]), CF(5) ), 4 ) ]
      gap> x:= last[1];;  x^2;
      IdentityMapping( AsField( NF(5,[ 1, 4 ]), CF(5) ) )

  ## 58.5 Gaussians

- `GaussianIntegers V`

  `GaussianIntegers` is the ring Z[√{−1}] of Gaussian integers. This is a subring of the cyclotomic field `GaussianRationals`, see GaussianRationals.

- `IsGaussianIntegers(` obj`) C`

  is the defining category for the domain `GaussianIntegers`.

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  GAP 4 manual
  October 2007