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

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 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.

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