## Generators for maximal ideals

Hello Professor Kim,

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

$m= X^3 - 2$

How do we make our decision to use

$X^3 - 2 = (X - 3)(X^2 + 3X - 1) \mod 5$

to find $P_5$ and $P_{25}$

How do we know not to use

$X^3 - 2 = (X - 3)(X^2 + 3X + 4) \mod 5$

or any other variation,e.g.,

$X^3 - 2 = (X - 3)(X^2 + 3X + 9) \mod 5$

Thanks

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 $F_5[x]$. So the question is, which lift to $Z[x]$ does one use to write down generators for the maximal ideals? And the answer is:

Any!

In your example, the point is that

$(5, \alpha^2 + 3\alpha - 1)=(5, \alpha^2 + 3\alpha + 4)=(5, \alpha^2 + 3\alpha + 9)$

in $O_K=Z[\alpha]$, where $\alpha=2^{1/3}$. The elements are different, but the ideals listed are the same, because 5 is already included in each of them. For example,

$\alpha^2 + 3\alpha + 4=(\alpha^2 + 3\alpha -1)+5$

and hence,

$\alpha^2 + 3\alpha + 4\in (5, \alpha^2 + 3\alpha - 1).$

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

$(p, f(\alpha))$

so that

$(p, f(\alpha))=(p,g(\alpha))$

whenever $f(x)\equiv g(x) \mod p$.