DefaultField( z,.. )
DefaultField( list )
In the first form DefaultField
returns the default field that contains
all the elements z,.. etc. In the second form DefaultField
returns
the default field that contains all the elements in the list list. If
any element is not an element of a field or the elements lie in no common
field an error is raised.
The field returned by DefaultField
need not be the smallest field in
which the elements lie. For example for elements from cyclotomic fields
DefaultField
may return the smallest cyclotomic field in which the
elements lie, which need not be the smallest field overall, because the
elements may in fact lie in a smaller number field which is not a
cyclotomic field.
For the exact definition of the default field of a certain type of elements read the chapter describing this type (see Finite Fields and Subfields of Cyclotomic Fields).
DefaultField
is used by Conjugates
, Norm
, Trace
, CharPol
, and
MinPol
(see Conjugates, Norm, Trace, CharPol, and MinPol) if
no explicit field is given.
gap> DefaultField( Z(4) ); GF(2^2) gap> DefaultField( E(9) ); CF(9) gap> DefaultField( [ Z(4), Z(9) ] ); Error, CharFFE: <z> must be a finite field element, vector, or matrix gap> DefaultField( [ E(4), E(9) ] ); CF(36)
Field
(see Field) differs from DefaultField
in that it returns the
smallest field in which the elements lie, while DefaultField
may return
a larger field if that makes sense.
GAP 3.4.4