.I just need a counter example for each of theses problems, they have been killing me for a few days now..

A) The gcd of

in is

, so the gcd is , up to a constant, and thus your answer is correct, though perhaps not the prettiest one.

B) If is a field in is a non constant polynomial having no roots in , then is irreducible in

has no roots over , but it is terribly NOT irreducible...

Tonio

These are both false, but Everything i try works for them...