GAP Manual: 15.8. GaloisGroup for Number Fields

The Galois automorphisms of the cyclotomic field Q_n are given by linear extension of the maps astk: e_n mapstoe_n^k with 1 leqk < n and Gcd( n, k ) = 1 (see GaloisCyc). Note that this action is not equal to exponentiation of cyclotomics, i.e., in general z^(astk) is different from 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 ) not= 1, the map e_n mapstoe_n^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

The Galois group Gal( Q_n, Q ) of the field extension Q_n/Q is isomorphic to the group (Z/nZ)^(ast) of prime residues modulo n, via the isomorphism

[ ccc (Z/nZ)^ast & rightarrow& Gal( Q_n, Q )
k & mapsto& ( z mapstoz^astk ) , ]

thus the Galois group of the field extension Q_n / L with L subseteqQ_n which is simply the factor group of Gal( Q_n, Q ) modulo the stabilizer of L, and the Galois group of L/L^(prime) which is the subgroup in this group that stabilizes L^(prime), are easily described in terms of (Z/nZ)^(ast) (Generators of (Z/nZ)^(ast) can be computed using GeneratorsPrimeResidues GeneratorsPrimeResidues.).

The Galois group of a field extension can be computed using GaloisGroup GaloisGroup:

    gap> f:= NF( [ EY(48) ] );
    NF(48,[ 1, 47 ])
    gap> g:= GaloisGroup( f );
    Group( NFAutomorphism( NF(48,[ 1, 47 ]) , 17 ), NFAutomorphism( NF(48,
    [ 1, 47 ]) , 11 ), NFAutomorphism( NF(48,[ 1, 47 ]) , 17 ) )
    gap> Size( g ); IsCyclic( g ); IsAbelian( g );
    8
    false
    true
    gap> f.base[1]; g.1; f.base[1] ^ g.1;
    E(24)-E(24)^11
    NFAutomorphism( NF(48,[ 1, 47 ]) , 17 )
    E(24)^17-E(24)^19
    gap> Operation( g, NormalBaseNumberField( f ), OnPoints );
    Group( (1,6)(2,4)(3,8)(5,7), (1,4,8,5)(2,3,7,6), (1,6)(2,4)(3,8)
    (5,7) )

The number field automorphism NFAutomorphism( F, k ) maps each element x of F to GaloisCyc( x, k ), see GaloisCyc.