Factorize with unknown in MAGMA

Question

Say I have a polynomial $f(x)$ over a finite field $\mathbb{F}_p$ such as $$f(x) = x^2 + \alpha x + \alpha^2$$ for some $\alpha \in \mathbb{F}_p$. Then we can express the zeroes $x_0, x_1$ in terms of $\alpha$. Can we do this in MAGMA as well? Somehow I cannot get it to factorize in terms of such an unknown $\alpha$.

Answer 1

Yes, this can be done. The point is that Magma does not work with "unknowns", but with function fields. Explicitly

F := GF(p);
K<a> := FunctionField(F);
P<x> := PolynomialRing(K);
f := x^2 + a*x + a^2;
Factorisation(f);

However, $f$ may not factor over $\mathbb{F}_p(\alpha)$. So in general you'd want to work with

F<u> := GF(p^2);
K<a> := FunctionField(F);
P<x> := PolynomialRing(K);
f := x^2 + a*x + a^2;
Factorisation(f);