Prove that is maximal in , where .
We will prove this in slightly more generality, beginning with a lemma.
Lemma: Let be a squarefree integer with mod 4. Then is a basis for over . Proof: Let . Then for some . Evidently, , where and . So . Suppose now that ; then and , so that , as desired.
Let be a squarefree integer with mod 4. Now the ring of integers in is . We claim that is maximal in .
To see this, note that . Now let , and note that mod . In particular, if mod 2, then mod , and if mod 2, then mod . We claim that these cosets are distinct. Indeed, if mod , then . In this case, by the lemma we have such that . But then and , so that , a contradiction. So . Now is a unital ring with having only two elements, so that ; that is, is a field, and thus is a maximal ideal.
The original problem follows since mod 4.