The Galois automorphisms of the cyclotomic field Q(en )
(see Chapter Cyclotomic Numbers) are given by
linear extension of the maps *k : en ® en k where
1 £ 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*k is different from zk:
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 ) ¹ 1, the map en ® en k is not a field automorphism but only a linear map.
gap> GaloisCyc( E(5)+E(5)^4, 5 ); GaloisCyc( ( E(5)+E(5)^4 )^2, 5 ); 2 -6
ANFAutomorphism( F, k ) F
Let F be an abelian number field F and k an integer.
If k is coprime to the conductor (see Conductor) of F then
ANFAutomorphism returns the automorphism of F defined as the linear
extension of the map that raises each root of unity in F to its k-th
power, otherwise an error is signalled.
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
The Galois group Gal( Q(en ), Q) of the field extension Q(en ) / Q is isomorphic to the group (Z/ n Z)* of prime residues modulo n, via the isomorphism from (Z/ n Z)* to Gal( Q(en ), Q) defined by k + n Z® ( z ® z*k ).
The Galois group of the field extension Q(en ) / L with
any abelian number field L Í Q(en ) is simply the
factor group of Gal( Qn , 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 (see GeneratorsPrimeResidues).
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) ]) gap> GeneratorsOfGroup( g ); [ ANFAutomorphism( CF(15), 11 ), ANFAutomorphism( CF(15), 7 ) ]
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GAP 4 manual