Abelian Number Fields
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Construction of Abelian Number Fields

Besides the usual construction using or (see ), abelian number fields consisting of cyclotomics can be created with and .

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Operations for Abelian Number Fields

For operations for elements of abelian number fields, e.g., or , see Chapter .

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) ]
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Integral Bases of Abelian Number Fields
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Galois Groups of Abelian Number Fields
abelian number fields
number fields
automorphism group
The field automorphisms of the cyclotomic field &QQ;_n (see Chapter ) are given by the linear maps *k on &QQ;_n that are defined by E(n)^{{*k}} = E(n)^k, where 1 \leq k < n and Gcd( n, k ) = 1 hold (see ). 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 &QQ;_n but only a &QQ;-linear map.

gap> GaloisCyc( E(5)+E(5)^4, 5 ); GaloisCyc( ( E(5)+E(5)^4 )^2, 5 ); 2 -6
The Galois group Gal( &QQ;_n, &QQ; ) of the field extension &QQ;_n / &QQ; is isomorphic to the group (&ZZ; / n &ZZ;)^{*} of prime residues modulo n, via the isomorphism (&ZZ; / n &ZZ;)^{*} \rightarrow Gal( &QQ;_n, &QQ; ) that is defined by k + n &ZZ; \mapsto ( z \mapsto z^{*k} ).

The Galois group of the field extension &QQ;_n / L with any abelian number field L \subseteq &QQ;_n is simply the factor group of Gal( &QQ;_n, &QQ; ) 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 (&ZZ; / n &ZZ;)^{*}. Generators of (&ZZ; / n &ZZ;)^{*} can be computed using .

In &GAP;, a field extension L / L' is given by the field object L with value L' (see ).

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) ) )
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Gaussians
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