## Norms of elements and principality

Dear Professor Kim,

Sorry to bombard you with these questions. I have come across a problem on your note ‘some principle ideals’. When we factorize modulo 3 we get we then associate these factors with the ideals and respectively. When we compute the norm of we do so by calculating the determinant of the matrix and find that the norm is in fact 9, so is a principle ideal. However, we could just have easily used or and in each case I get a different answer for the determinant. Have I made an error or is there a canonical form of sort that I should be aware of?

Thank you for your time.

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Reply:

First of all, I presume your etc. are etc. All the elements you mention do indeed belong to the ideal and can be used as generators *when used together with the element 3*. Indeed they are all all evaluations at of polynomials that are congruent to mod . However, this does not mean they are generators on their own. Of course different elements in an ideal will have different norms in general. However, an element is a generator *by itself* (making into a principal ideal), exactly when . Of course such a need not exist. I haven’t calculated the norms of the elements you mention, but if their norms come out larger than 9, it merely says they are not generators (again, by themselves), while is.

A thorny point that comes out of this discussion is that if you had initially presented the ideal as , for example, then it might have been harder to see that it is principal.

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## 2 Comments

Thanks for the feedback. So, if I understand you correctly, given a situation where we are checking for principality of an ideal, but we don’t find a generator with our first presentation, our only option would be to exhaust all the other possibilities?

`Exhaust other possibilities’ sounds exhausting. But recall that the equation

for an unknown element in our ring is not too difficult to analyze, at least in many examples. If there are no solutions at all, then a fortiori there are no solutions in our ideal. So the ideal would not be principal. However, if there is a solution, you would still need to check if there is a solution in the ideal. This can be somewhat trickier.