Show that and are relatively prime as algebraic integers in , but that there do not exist algebraic integers such that .
Let be an arbitrary integer in . Since mod 4, and are integers. Note that . If this element has norm 3, then taking this equation mod 5 we have . Note, however, that the squares mod 5 are 0, 1, and 4. In particular, no algebraic integer in has norm 3.
Now and ; if these elements have a factorization, then some factor must be a unit. Thus both and are irreducible as integers in . By Theorem 7.7 in TAN, the units in this ring are precisely , so that 3 and are not associates.
If and , then since , , and thus by Lemma 7.3 in TAN is a unit. That is, every common divisor of and is a unit, so that these elements are relatively prime.
Suppose now that there are integers such that, with and , we have .Comparing coefficients, we have and . Mod 3, these equations reduce to and , a contradiction. So no such and exist.