Dear GAP Forum,
To answer a variety of mails by Igor Schein:
Hi, I'm a GAP novice. I was wondering how I can specify a number
field associated with a given polynomial, i.e. a field obtained by
adjoining roots of a polynomial in Q[x] to Q.
There is a command `AlgebraicExtension' that will construct such an extension
field. For example in GAP4, you could use:
gap> x:=Indeterminate(Rationals,"x"); x gap> p:=x^3+x-27; -27+x+x^3 gap> e:=AlgebraicExtension(Rationals,p); <field in characteristic 0> gap> a:=PrimitiveElement(e); (a) gap> a^7*19; (513+13832*a-1026*a^2) gap> Value(p,a); !0 # denotes 0 naturally embedded in the extension
As you found out yourself, there is not yet a method for
``RootOfDefiningPolynomial''. We will add one in future versions.
At the Moment, `PrimitiveElement' for extensions constructed
via `AlgebraicExtension' will give you a root of the defining polynomial.
> I wonder what's included in GAP's definition of *standard finite fields*.
Well, firstly they have to be finite fields, that is fields with finitely
many elements. The field you were defining is infinite (though of finite
index over the rationals).
A *standard finite field * (the way we use the term in the library) is a
finite field whose elements are implemented in the kernel (via
Zech-logarithms) and are displayed as `Z(p)^e'.
I hope this is of help,