Checking principality again


I was wondering how you work out when an ideal is principal. I understand that a principal ideal is generated by a single element, but how do you work this out. For example in the remarks, you mentioned (5,2^{1/3}-3) was the principal ideal (2(2^{2/3})+2^{1/3}+2). But where does this come from? (5,2^{1/3}-3) is elements a5+b(2^{1/3}-3) with a,b \in Z[2^{1/3}]. I tried to substitute x+y2^{1/3} for a and b, but I’m not sure if this is right, and if it is, where do the specific numbers come from. Thanks for your help.


Checking for principality is not always easy, although it is sometimes. For example, when you factorize the ideal (2) in the ring Z[\alpha] where \alpha is a root of x^3-4x+2, you’ll see that (2)=P_2^2 where P_2=(2, \alpha). But clearly, 2=4\alpha-\alpha^3, so that \alpha by itself is a generator of P_2.

One general rule that comes up often is that for a principal ideal I=(a), we have N(I)=|N(a)|. From this, it is easy to deduce that

I is principal if and only if there is an element a \in I such that |N(a)|=I.

You will find a number of examples of this flavor in the note `Some principal ideals’ . The example you mention has a misprint. The actual generator for the ideal is 2^{2/3}+1. This is not so straightforward, and is worked out on page two of `Some sample class groups’.


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