A computational question: suppose you have a number field, $F$, and the minimal polynomial, $f$, of a primitive element, $x$, of $F$, so that you can represent elements of $F$ as rational polynomials in $x$ between which computations are done modulo $f$. Does taking the GCD's of polynomials whose coefficients are in $F$ using the basic Euclidean algorithm suffer from coefficient explosion?

algebra rules!