## Generators for maximal ideals

Hello Professor Kim,

I’ve found myself stuck on finding maximal ideals when finding class groups. for example:

How do we make our decision to use

to find and

How do we know not to use

or any other variation,e.g.,

Thanks

Reply:

This is a good question! Seeing it, I can see you’re reviewing seriously. Note first of all that all the degree two polynomials you’ve listed are the same in . So the question is, which lift to does one use to write down generators for the maximal ideals? And the answer is:

Any!

In your example, the point is that

in , where . The elements are different, but the ideals listed are the same, because 5 is already included in each of them. For example,

and hence,

The maximal ideals occurring in Dedekind’s theorem all look like

so that

whenever .

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