> < ^ Date: Mon, 25 Jan 1999 10:35:35 +0000 (GMT)
> < ^ From: Alexander Hulpke <hulpke@math.colostate.edu >
< ^ Subject: Re: number fields

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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