‣ GaloisGroup ( F ) | ( method ) |
The Galois group of the field extension is isomorphic to the group of prime residues modulo , via the isomorphism that is defined by .
The Galois group of the field extension with any abelian number field is simply the factor group of modulo the stabilizer of , and the Galois group of , with an abelian number field contained in , is the subgroup in this group that stabilizes . These groups are easily described in terms of . Generators of can be computed using GeneratorsPrimeResidues
(15.2-4).
In GAP, a field extension is given by the field object with LeftActingDomain
(57.1-11) value (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 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) ) )