Determine which of the following ideals of are equivalent: , , , and .
First we argue that is principal. Indeed, . Now , so that . Indeed, and , and .
In particular, .
Now we claim that and are both principal. Indeed, . Since and , we have . So . Similarly, , so that . Now , so that .
Finally, we claim that is not principal. If , then and , so that divides both 4 and 6. So . Note, however, that no element of has norm 2 since the equation has no solutions in . If , then we have . But then , so that , which is impossible mod 2. So is not principal.
In summary, we have , , and .