60.4-1 GaloisGroup
‣ GaloisGroup( F )( method )

The Galois group G a l ( ? n , ? ) 𝐺 𝑎 𝑙 subscript ? 𝑛 ? Gal(?_{n},?) of the field extension ? n / ? subscript ? 𝑛 ? ?_{n}/? is isomorphic to the group ( ? / n ? ) * superscript ? 𝑛 ? (?/n?)^{*} of prime residues modulo n 𝑛 n , via the isomorphism ( ? / n ? ) * G a l ( ? n , ? ) superscript ? 𝑛 ? 𝐺 𝑎 𝑙 subscript ? 𝑛 ? (?/n?)^{*}\rightarrow Gal(?_{n},?) that is defined by k + n ? ( z z * k ) fragments k n ? maps-to fragments ( z maps-to superscript 𝑧 absent 𝑘 ) k+n?\mapsto(z\mapsto z^{*k}) .

The Galois group of the field extension ? n / L subscript ? 𝑛 𝐿 ?_{n}/L with any abelian number field L ? n 𝐿 subscript ? 𝑛 L\subseteq?_{n} is simply the factor group of G a l ( ? n , ? ) 𝐺 𝑎 𝑙 subscript ? 𝑛 ? Gal(?_{n},?) modulo the stabilizer of L 𝐿 L , and the Galois group of L / L 𝐿 superscript 𝐿 L/L^{\prime} , with L superscript 𝐿 L^{\prime} an abelian number field contained in L 𝐿 L , is the subgroup in this group that stabilizes L superscript 𝐿 L^{\prime} . These groups are easily described in terms of ( ? / n ? ) * superscript ? 𝑛 ? (?/n?)^{*} . Generators of ( ? / n ? ) * superscript ? 𝑛 ? (?/n?)^{*} can be computed using GeneratorsPrimeResidues (15.2-4).

In GAP, a field extension L / L 𝐿 superscript 𝐿 L/L^{\prime} is given by the field object L 𝐿 L with LeftActingDomain (57.1-11) value L superscript 𝐿 L^{\prime} (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) ) )