% This file was created automatically from fldabnum.msk. % DO NOT EDIT! %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %A fldabnum.msk GAP documentation Thomas Breuer %% %A @(#)$Id: fldabnum.msk,v 1.12.2.1 2006/09/16 19:02:49 jjm Exp $ %% %Y (C) 1998 School Math and Comp. Sci., University of St. Andrews, Scotland %Y Copyright (C) 2002 The GAP Group %% \Chapter{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~"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, $\Q_n$ 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, % section about Gaussians here? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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 $\Q_n$. 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.~"Integral Bases of Abelian Number Fields"). \indextt{CF} `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. \beginexample 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) ) \endexample \>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. \indextt{NF} `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. \beginexample 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 \endexample \>`GaussianRationals' V \>IsGaussianRationals( <obj> ) C `GaussianRationals' is the field $\Q_4 = \Q(\sqrt{-1})$ 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".) \beginexample gap> CF(4) = GaussianRationals; true gap> Sqrt(-1) in GaussianRationals; true \endexample % factoring of elements in GaussianRationals works? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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". \index{cyclotomics!DefaultField} 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"). \beginexample 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 ]) \endexample \index{polynomials over abelian number fields!Factors} 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. \beginexample 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) ] \endexample \>IsNumberField( <F> ) P \index{number field} returns `true' if the field <F> is a finite dimensional extension of a prime field in characteristic zero, and `false' otherwise. \>IsAbelianNumberField( <F> ) P \index{abelian number field} 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. \beginexample 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 \endexample \>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>. \beginexample 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 ] \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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 $\Q_n$ 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' (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"). \index{cyclotomic fields!CanonicalBasis} 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( <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>. \index{abelian number fields!CanonicalBasis} 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 $\Q_m$, 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>. \beginexample 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 ] \endexample \>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 $\Q_n$, viewed as a vector space over $\Q_m$. This basis is defined as follows. Let $P$ denote the set of prime divisors of <n>, $<n> = \prod_{p\in P} p^{\nu_p}$, and $<m> = \prod_{p\in P} p^{\mu_p}$ with $\mu_p \leq \nu_p$. Let $e_n = `E(<n>)'$, 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 $$ B_{n,m} = \bigotimes_{p\in P} \bigotimes_{k=\mu_p}^{\nu_p-1} \{ e_{p^{k+1}}^j\}_{j\in J_{k,p}} {\rm\ \ where\ \ } J_{k,p} = \left\{ \matrix{ \{ 0 \} & ; & k=0, p=2 \cr \{ 0, 1 \} & ; & k > 0, p=2 \cr \{ 1, \ldots, p-1 \} & ; & k = 0, p\not= 2 \cr \{ -\frac{p-1}{2}, \ldots, \frac{p-1}{2} \} & ; & k > 0, p\not= 2 \cr } \right. $$ $B_{n,1}$ is equal to the basis of $\Q_n$ over the rationals which is introduced in~\cite{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"), it is a normal basis for squarefree $n$ and closed under complex conjugation for odd $n$. *Note:* For $<n> \equiv 2 \pmod 4$, 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 )'. \beginexample 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 ] \endexample \>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 $[ b_1, b_2, \ldots, b_k ]$ 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( <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~\cite{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 $b_i$ 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>. \beginexample 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 ] ] \endexample The first two results describe two bases of the field $\Q_3(\sqrt{6})$, the third result describes a normal basis of $\Q_3(\sqrt{5})$. %T missing: `IsIntegralBasis', `NormalBasis', `IsNormalBasis', %T rings of integers in abelian number fields %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Galois Groups of Abelian Number Fields} \atindex{abelian number fields!Galois group}% {@abelian number fields!Galois group} \atindex{number fields!Galois group}{@number fields!Galois group} \index{automorphism group!of number fields} The field automorphisms of the cyclotomic field $\Q_n$ (see Chapter~"Cyclotomic Numbers") are given by the linear maps $\ast k$ on $\Q_n$ that are defined by $`E'(n)^{\ast k} = `E'(n)^k$, where $1 \leq k \< n$ and $`Gcd'( n, k ) = 1$ hold (see~"GaloisCyc"). Note that this action is *not* equal to exponentiation of cyclotomics, i.e., for general cyclotomics $z$, $z^{\ast k}$ is different from $z^k$. (In {\GAP}, the image of a cyclotomic $z$ under $\ast k$ can be computed as $`GaloisCyc'( z, k )$.) \beginexample 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 \endexample For $`Gcd'( n, k ) \not= 1$, the map $`E'(n) \mapsto `E'(n)^k$ does *not* define a field automorphism of $\Q_n$ but only a $\Q$-linear map. \beginexample gap> GaloisCyc( E(5)+E(5)^4, 5 ); GaloisCyc( ( E(5)+E(5)^4 )^2, 5 ); 2 -6 \endexample \>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. \beginexample 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 \endexample The Galois group $Gal( \Q_n, \Q )$ of the field extension $\Q_n / \Q$ is isomorphic to the group $(\Z / n \Z)^{\ast}$ of prime residues modulo $n$, via the isomorphism $(\Z / n \Z)^{\ast} \rightarrow Gal( \Q_n, \Q )$ that is defined by $k + n \Z \mapsto ( z \mapsto z^{\ast k} )$. 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 $L / L^{\prime}$, with $L^{\prime}$ an abelian number field contained in $L$, is the subgroup in this group that stabilizes $L^{\prime}$. These groups are easily described in terms of $(\Z / n \Z)^{\ast}$. Generators of $(\Z / n \Z)^{\ast}$ can be computed using `GeneratorsPrimeResidues' (see~"GeneratorsPrimeResidues"). In {\GAP}, a field extension $L / L^{\prime}$ is given by the field object $L$ with `LeftActingDomain' value $L^{\prime}$ (see~"Integral Bases of Abelian Number Fields"). \beginexample 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) ]) \endexample The following example shows Galois groups of a cyclotomic field and of a proper subfield that is not a cyclotomic field. \beginexample 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 \endexample The following example shows the Galois group of a cyclotomic field over a non-cyclotomic field. \beginexample 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) ) ) \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Gaussians} \>`GaussianIntegers' V `GaussianIntegers' is the ring $\Z[\sqrt{-1}]$ 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'. % Gcd and Euclidean... for the rings of integers in CF(4) and CF(3) ! %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E