Suppose is an irreducible element in an algebraic integer ring . Must be prime as an ideal in ?
Recall that is irreducible in , where , since no element in this ring has norm 3. Moreover, , as we show. If to the contrary we have , then comparing coefficients we have and , so that for some , a contradiction. Using this previous exercise, we have . In particular, is irreducible, but is not maximal and thus not prime.
So it need not be the case that is prime if is irreducible.