# 15.8 GaloisGroup for Number Fields

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
April 1997