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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%A fields.msk GAP documentation Thomas Breuer
%%
%A @(#)$Id: fields.msk,v 1.17.2.2 2006/09/16 19:02:49 jjm Exp $
%%
%Y (C) 1998 School Math and Comp. Sci., University of St. Andrews, Scotland
%Y Copyright (C) 2002 The GAP Group
%%
\Chapter{Fields and Division Rings}
\index{fields}
\index{division rings}
\FileHeader{field}[1]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Generating Fields}
\Declaration{IsDivisionRing}
\Declaration{IsField}
\beginexample
gap> IsField( GaloisField(16) ); # the field with 16 elements
true
gap> IsField( Rationals ); # the field of rationals
true
gap> q:= QuaternionAlgebra( Rationals );; # a noncommutative division ring
gap> IsField( q ); IsDivisionRing( q );
false
true
gap> mat:= [ [ 1 ] ];; a:= Algebra( Rationals, [ mat ] );;
gap> IsDivisionRing( a ); # an algebra not constructed as a division ring
false
\endexample
\Declaration{Field}
\Declaration{DefaultField}
\beginexample
gap> Field( Z(4) ); Field( [ Z(4), Z(8) ] ); # finite fields
GF(2^2)
GF(2^6)
gap> Field( E(9) ); Field( CF(4), [ E(9) ] ); # abelian number fields
CF(9)
AsField( GaussianRationals, CF(36) )
gap> f1:= Field( EB(5) ); f2:= DefaultField( EB(5) );
NF(5,[ 1, 4 ])
CF(5)
gap> f1 = f2; IsSubset( f2, f1 );
false
true
\endexample
\Declaration{DefaultFieldByGenerators}
\Declaration{GeneratorsOfDivisionRing}
\Declaration{GeneratorsOfField}
\Declaration{DivisionRingByGenerators}
\Declaration{AsDivisionRing}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Subfields of Fields}
\Declaration{Subfield}
\Declaration{FieldOverItselfByGenerators}
\Declaration{PrimitiveElement}
\Declaration{PrimeField}
\Declaration{IsPrimeField}
\Declaration{DegreeOverPrimeField}
\Declaration{DefiningPolynomial}
\Declaration{RootOfDefiningPolynomial}
\Declaration{FieldExtension}
\Declaration{Subfields}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Galois Action}
\FileHeader{field}[2]
\indextt{IsFieldControlledByGaloisGroup}
\Declaration{GaloisGroup}[field]!{of field}
\beginexample
gap> g:= GaloisGroup( AsField( GF(2^2), GF(2^12) ) );;
gap> Size( g ); IsCyclic( g );
6
true
gap> h:= GaloisGroup( CF(60) );;
gap> Size( h ); IsAbelian( h );
16
true
\endexample
\>MinimalPolynomial( <F>, <z>[, <ind>] )!{over a field} O
returns the minimal polynomial of <z> over the field <F>.
This is a generator of the ideal in $<F>[x]$ of all polynomials
which vanish on <z>.
(This definition is consistent with the general definition of
`MinimalPolynomial' for rings, see~"MinimalPolynomial".)
\beginexample
gap> MinimalPolynomial( Rationals, E(8) );
x_1^4+1
gap> MinimalPolynomial( CF(4), E(8) );
x_1^2+(-E(4))
gap> MinimalPolynomial( CF(8), E(8) );
x_1+(-E(8))
\endexample
\Declaration{TracePolynomial}
\index{characteristic polynomial!for field elements}
\beginexample
gap> TracePolynomial( CF(8), Rationals, E(8) );
x_1^4+1
gap> TracePolynomial( CF(16), Rationals, E(8) );
x_1^8+2*x_1^4+1
\endexample
\Declaration{Norm}
\Declaration{Trace}!{for field elements}
\Declaration{Conjugates}
\beginexample
gap> Norm( E(8) ); Norm( CF(8), E(8) );
1
1
gap> Norm( CF(8), CF(4), E(8) );
-E(4)
gap> Norm( AsField( CF(4), CF(8) ), E(8) );
-E(4)
gap> Trace( E(8) ); Trace( CF(8), CF(8), E(8) );
0
E(8)
gap> Conjugates( CF(8), E(8) );
[ E(8), E(8)^3, -E(8), -E(8)^3 ]
gap> Conjugates( CF(8), CF(4), E(8) );
[ E(8), -E(8) ]
gap> Conjugates( CF(16), E(8) );
[ E(8), E(8)^3, -E(8), -E(8)^3, E(8), E(8)^3, -E(8), -E(8)^3 ]
\endexample
\FileHeader{field}[3]
\Declaration{NormalBase}
\beginexample
gap> NormalBase( CF(5) );
[ -E(5), -E(5)^2, -E(5)^3, -E(5)^4 ]
gap> NormalBase( CF(4) );
[ 1/2-1/2*E(4), 1/2+1/2*E(4) ]
gap> NormalBase( GF(3^6) );
[ Z(3^6)^2, Z(3^6)^6, Z(3^6)^18, Z(3^6)^54, Z(3^6)^162, Z(3^6)^486 ]
gap> NormalBase( GF( GF(8), 2 ) );
[ Z(2^6), Z(2^6)^8 ]
\endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\Section{Field Homomorphisms}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%E
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