## Checking principality again

Hi,

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 was the principal ideal . But where does this come from? is elements with . I tried to substitute 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.

Reply:

Checking for principality is not always easy, although it is sometimes. For example, when you factorize the ideal (2) in the ring where is a root of , you’ll see that where . But clearly, , so that by itself is a generator of .

One general rule that comes up often is that for a principal ideal , we have . From this, it is easy to deduce that

is principal if and only if there is an element such that .

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 . This is not so straightforward, and is worked out on page two of `Some sample class groups’.

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