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60 Abelian Number Fields

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 18). Note that abelian number fields can also be constructed with the more general `AlgebraicExtension`

(67.1-1), a discussion of advantages and disadvantages can be found in 18.6. The functions described in this chapter have been developed for fields whose elements are in the filter `IsCyclotomic`

(18.1-3), they may or may not work well for abelian number fields consisting of other kinds of elements.

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

In 60.1, constructors for abelian number fields are described, 60.2 introduces operations for abelian number fields, 60.3 deals with the vector space structure of abelian number fields, and 60.4 describes field automorphisms of abelian number fields,

Besides the usual construction using `Field`

(58.1-3) or `DefaultField`

(18.1-16) (see `DefaultField`

(18.1-16)), abelian number fields consisting of cyclotomics can be created with `CyclotomicField`

(60.1-1) and `AbelianNumberField`

(60.1-2).

`‣ CyclotomicField` ( [subfield, ]n ) | ( function ) |

`‣ CyclotomicField` ( [subfield, ]gens ) | ( function ) |

`‣ CF` ( [subfield, ]n ) | ( function ) |

`‣ CF` ( [subfield, ]gens ) | ( function ) |

The first version creates the `n`-th cyclotomic field \(ℚ_n\). 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. 60.3).

`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 ) | ( function ) |

`‣ NF` ( n, stab ) | ( function ) |

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`

(60.2-5)), in the `n`-th cyclotomic field. `AbelianNumberField`

is mainly thought for internal use and for printing fields in a standard way; `Field`

(58.1-3) (cf. also 60.2) 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` | ( global variable ) |

`‣ IsGaussianRationals` ( obj ) | ( category ) |

`GaussianRationals`

is the field \(ℚ_4 = ℚ(\sqrt{{-1}})\) of Gaussian rationals, as a set of cyclotomic numbers, see Chapter 18 for basic operations. This field can also be obtained as `CF(4)`

(see `CyclotomicField`

(60.1-1)).

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`

(17.1-1).)

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

For operations for elements of abelian number fields, e.g., `Conductor`

(18.1-7) or `ComplexConjugate`

(18.5-2), see Chapter 18.

`‣ Factors` ( F ) | ( method ) |

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 ) | ( property ) |

returns `true`

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

otherwise.

`‣ IsAbelianNumberField` ( F ) | ( property ) |

returns `true`

if the field `F` is a number field (see `IsNumberField`

(60.2-2)) that is a Galois extension of the prime field, with abelian Galois group (see `GaloisGroup`

(58.3-1)).

`‣ IsCyclotomicField` ( F ) | ( property ) |

returns `true`

if the field `F` is a *cyclotomic field*, i.e., an abelian number field (see `IsAbelianNumberField`

(60.2-3)) 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 ) | ( attribute ) |

Let `F` be an abelian number field (see `IsAbelianNumberField`

(60.2-3)) with conductor \(n\), say. (This means that the \(n\)-th cyclotomic field is the smallest cyclotomic field containing `F`, see `Conductor`

(18.1-7).) `GaloisStabilizer`

returns the set of all those integers \(k\) in the range \([ 1 .. n ]\) 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 ]

Each abelian number field is naturally a vector space over \(ℚ\). Moreover, if the abelian number field \(F\) contains the \(n\)-th cyclotomic field \(ℚ_n\) then \(F\) is a vector space over \(ℚ_n\). 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`

(57.1-11).

The return values of `NF`

(60.1-2) and of the one argument versions of `CF`

(60.1-1) represent vector spaces over \(ℚ\), and the return values of the two argument version of `CF`

(60.1-1) 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`

(58.1-9).

Let `F` be the cyclotomic field \(ℚ_n\), represented as a vector space over the subfield `S`. If `S` is the cyclotomic field \(ℚ_m\), with \(m\) a divisor of \(n\), then `CanonicalBasis( `

returns the Zumbroich basis of `F` )`F` relative to `S`, which consists of the roots of unity `E(`

^`n`)`i` where `i` is an element of the list `ZumbroichBase( `

(see `n`, `m` )`ZumbroichBase`

(60.3-1)). If `S` is an abelian number field that is not a cyclotomic field then `CanonicalBasis( `

returns a normal `F` )`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( `

, with conductor `n`, `stab` )`n`, that is itself not a cyclotomic field, represented as a vector space over the subfield `S`. If `S` is the cyclotomic field \(ℚ_m\), with \(m\) a divisor of \(n\), then `CanonicalBasis( `

returns the Lenstra basis of `F` )`F` relative to `S` that consists of the sums of roots of unity described by `LenstraBase( `

(see `n`, `stab`, `stab`, `m` )`LenstraBase`

(60.3-2)). If `S` is an abelian number field that is not a cyclotomic field then `CanonicalBasis( `

returns a normal `F` )`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 ) | ( function ) |

Let `n` and `m` be positive integers, such that `m` divides `n`. `ZumbroichBase`

returns the set of exponents \(i\) for which `E(`

\(i\) belongs to the (generalized) Zumbroich basis of the cyclotomic field \(ℚ_n\), viewed as a vector space over \(ℚ_m\).`n`)^

This basis is defined as follows. Let \(P\) denote the set of prime divisors of `n`, \(\textit{n} = \prod_{{p \in P}} p^{{\nu_p}}\), and \(\textit{m} = \prod_{{p \in P}} p^{{\mu_p}}\) with \(\mu_p \leq \nu_p\). Let \(e_l =\) `E`

\((l)\) for any positive integer \(l\), and \(\{ e_{{n_1}}^j \}_{{j \in J}} \otimes \{ e_{{n_2}}^k \}_{{k \in K}} = \{ e_{{n_1}}^j \cdot e_{{n_2}}^k \}_{{j \in J, k \in K}}\).

Then the basis is

\[ B_{{n,m}} = \bigotimes_{{p \in P}} \bigotimes_{{k = \mu_p}}^{{\nu_p-1}} \{ e_{{p^{{k+1}}}}^j \}_{{j \in J_{{k,p}}}} \]

where \(J_{{k,p}} =\)

\(\{ 0 \}\) | ; | \(k = 0, p = 2\) |

\(\{ 0, 1 \}\) | ; | \(k > 0, p = 2\) |

\(\{ 1, \ldots, p-1 \}\) | ; | \(k = 0, p \neq 2\) |

\(\{ -(p-1)/2, \ldots, (p-1)/2 \}\) | ; | \(k > 0, p \neq 2\) |

\(B_{{n,1}}\) is equal to the basis of \(ℚ_n\) 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.)

\(B_{{n,m}}\) consists of roots of unity, it is an integral basis (that is, exactly the integral elements in \(ℚ_n\) have integral coefficients w.r.t. \(B_{{n,m}}\), cf. `IsIntegralCyclotomic`

(18.1-4)), it is a normal basis for squarefree \(n\) and closed under complex conjugation for odd \(n\).

*Note:* For \(\textit{n} \equiv 2 \pmod 4\), we have `ZumbroichBase(`

and `n`, 1) = 2 * ZumbroichBase(`n`/2, 1)`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 ) | ( function ) |

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`

(60.2-5)), and `super` be a list representing a supergroup of the group given by `stabilizer`.

`LenstraBase`

returns a list \([ b_1, b_2, \ldots, b_k ]\) of lists, each \(b_i\) consisting of integers such that the elements \(\sum_{{j \in b_i}} \)`E(n)`

\(^j\) form a basis of the abelian number field `NF( `

, as a vector space over the `n`, `stabilizer` )`m`-th cyclotomic field (see `AbelianNumberField`

(60.1-2)).

This basis is an integral basis, that is, exactly the integral elements in `NF( `

have integral coefficients. (For details about this basis, see [Bre97].)`n`, `stabilizer` )

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 \(b_i\) are in general not sets, since for

, the first entry is always an element of `stabilizer` = `super``ZumbroichBase( `

; this property is used by `n`, `m` )`NF`

(60.1-2) and `Coefficients`

(61.6-3) (see 60.3).

`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 \(ℚ_3(\sqrt{{6}})\), the third result describes a normal basis of \(ℚ_3(\sqrt{{5}})\).

The field automorphisms of the cyclotomic field \(ℚ_n\) (see Chapter 18) are given by the linear maps \(*k\) on \(ℚ_n\) that are defined by `E`

\((n)^{{*k}} = \)`E`

\((n)^k\), where \(1 \leq k < n\) and `Gcd`

\(( n, k ) = 1\) hold (see `GaloisCyc`

(18.5-1)). Note that this action is *not* equal to exponentiation of cyclotomics, i.e., for general cyclotomics \(z\), \(z^{{*k}}\) is different from \(z^k\).

(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`

\(( n, k ) \neq 1\), the map `E`

\((n) \mapsto\) `E`

\((n)^k\) does *not* define a field automorphism of \(ℚ_n\) but only a \(ℚ\)-linear map.

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

`‣ GaloisGroup` ( F ) | ( method ) |

The Galois group \(Gal( ℚ_n, ℚ )\) of the field extension \(ℚ_n / ℚ\) is isomorphic to the group \((ℤ / n ℤ)^{*}\) of prime residues modulo \(n\), via the isomorphism \((ℤ / n ℤ)^{*} \rightarrow Gal( ℚ_n, ℚ )\) that is defined by \(k + n ℤ \mapsto ( z \mapsto z^{*k} )\).

The Galois group of the field extension \(ℚ_n / L\) with any abelian number field \(L \subseteq ℚ_n\) is simply the factor group of \(Gal( ℚ_n, ℚ )\) 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`

(15.2-4).

In **GAP**, a field extension \(L / L'\) is given by the field object \(L\) with `LeftActingDomain`

(57.1-11) value \(L'\) (see 60.3).

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 of size 2 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) ) )

`‣ ANFAutomorphism` ( F, k ) | ( function ) |

Let `F` be an abelian number field and `k` be an integer that is coprime to the conductor (see `Conductor`

(18.1-7)) 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

`‣ GaussianIntegers` | ( global variable ) |

`GaussianIntegers`

is the ring \(ℤ[\sqrt{{-1}}]\) of Gaussian integers. This is a subring of the cyclotomic field `GaussianRationals`

(60.1-3).

`‣ IsGaussianIntegers` ( obj ) | ( category ) |

is the defining category for the domain `GaussianIntegers`

(60.5-1).

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