How to construct the isomorphism on GAP?

Question

I would like to ask for some help regarding the syntax on GAP GAP syntax

(in short, I have $\tau$ a root of $x^2-x-1$ and want to map from $Z[\tau]/\pi$ (with $\pi$ being a given prime ideal) to the appropriate finite field.

Answer 1

As a group theory system, GAP has not much functionality for ring homomorphisms. There is basically no built-in functionality for what you want to do, you'd have to

1) Determine the prime $p\in Z$ that lies below $\pi$

2) Determine the minimal polynomial of $\tau$.

gap> tau:=(1+ER(5))/2;
-E(5)^2-E(5)^3
gap> pol:=MinimalPolynomial(Rationals,tau);
x_1^2-x_1-1

3) Factor the corresponding polynomial over the field with $p$ elements. E.g. for $p=7$:

gap> Factors(PolynomialModP(pol,7));
[ x_1^2-x_1-Z(7)^0 ]

The (smallest, but often they are the same) degree of a factor is the exponent $e$ such that the finite field is the one with $p^3$ element.

At the moment GAP has no functionality for the actual homomorphism but in -- you would have to write it yourself (by getting the coefficients in the cyclotomic field and forming the corresponding element of the finite field. You might want to talk with your advisor about what this means concretely).

This will be somewhat complicated code, and I would not recommend this problem to a complete beginner.