DefaultField( z,.. )
DefaultField( list )
In the first form
DefaultField returns the default field that contains
all the elements z,.. etc. In the second form
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
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
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.
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