In this case, simply computing the GCD may require "non-obvious" intuition, since the polynomials do not act over the real numbers, but the algebraic closure of

In this case, simply computing the GCD may require "non-obvious" intuition, since the polynomials do not act over the real numbers, but the algebraic closure of an unknown finite field. But I wouldn't bet my data on it, at least, not until I am a lot more confident in my understanding of abstract algebra.