### Accuracy versus precision of algebraic number calculations

Hi everyone - another very basic question from me ...

I am doing some calculations of absolute norms of determinants of matrices whose entries come from cyclotomic fields. The (rational) numbers which are output are sensible but occasionally have massive prime numbers as factors which I was not expecting. My question is whether I can rely upon such numbers when they are output by SAGE's calculations inside a specified number field, or whether somewhere along the way some imprecision may have been introduced which results in a "distorted" prime number being a factor of the output.

Another stylized way of asking the same thing: is it possible that the answer to some question involving a small prime p might actually contain a factor of p^100, but nevertheless because of accumulated rounding errors etc I have ended up with p^100-2 which happens to be prime? Or does SAGE "know" only to output perfect answers involving algebraic number fields, even when the heights involved are that big?

Many thanks