The Galois automorphisms of the cyclotomic field Q_n are given by
linear extension of the maps ast k: e_n mapsto e_n^k with
1 leq k < n and Gcd( n, k ) = 1 (see GaloisCyc).
Note that this action is not equal to exponentiation of cyclotomics,
i.e., in general z^{ast k} 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 mapsto e_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
[ beginarrayccc (Z/nZ)^ast & rightarrow & Gal( Q_n, Q )
k & mapsto & ( z mapsto z^ast k )
endarray , ]
thus the Galois group of the field extension Q_n / L with
L subseteq Q_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.
GAP 3.4.4