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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.

He continued:

> 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,

Alexander Hulpke

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