Recall that if is an algebraic number field with ring of integers and if is an ideal, then . Compute for each of the following ideals in their respective field : (1) in with , (2) in with , and (3) in with .
- We claim that in . Suppose in . In particular, , so for some . So with . Suppose and consider . If (in ) is arbitrary, then . So , and thus .
- We claim that in . Suppose in . In particular, , so that . Adding, we have , so that for some , and so . Now , so that for some . Now . Note that mod 2. So we have where and mod 2, as desired. Suppose such that mod 2 and consider . Note that , since mod 2. Thus for all , so we have , and thus .
- We claim that in . Suppose . In particular, so that (comparing coefficients) and for some . Moreover, we have , so that (comparing coefficients) . Say , so that . Mod 3, we have , as desired. Consider where mod 3. Now and since mod 3 and mod 3. So for all , and thus .