Find such that in . Find such that in .
Note that . Using Theorem 9.3, it suffices to find an element and an ideal such that and . To that end, note that factors into prime ideals as , with distinct factors , , and . Let , , and . Now let , , and . Evidently, , , and , and . Following the proof of Theorem 9.3, let . Then . Indeed, we have , , and .
We first need to find an ideal such that . In a previous exercise we found that is such an ideal. Next we wish to find the prime factorization of . To this end, we note that (since ).
Claim: is proper. Proof of claim: Suppose to the contrary that this ideal contains 1. THen we have , so that and . This leads to a contradiction mod 3, so that is proper.
Claim: is maximal. Proof of claim: Let , and say and where and . Mod , we have . Now mod . Thus and . In particular, . Now since this ideal is proper, and so also . If , then , a contradiction. So is a field, and thus is maximal.
We can show very similarly that is proper and also maximal. Here we can show that .
Thus we have the prime factorization of , with distinct factors and . Let and . We see that and .
Next we find and such that . By our proof of Theorem 9.2, it suffices to exhibit , and likewise for . Noting that , we see that and are such elements. Let . By the proof of Theorem 9.3 in TAN, .
We can verify this by noting that , , , and .