I am trying to factor the ideals (7) and (35) into products of prime ideals in the ring of integers of $K:= \mathbb{Q}(\sqrt[3]{2}).$
Well, I know, that the ring of integers of $K$ is $ O_K =\mathbb{Z}[\sqrt[3]{2}]$ (see: Easy way to show that $\mathbb{Z}[\sqrt[3]{2}]$ is the ring of integers of $\mathbb{Q}[\sqrt[3]{2}]$).
I am lacking a lot of theory and I do not know how to start.
$(35) = (7) (5)$, so I have to find factorizations of (7) and (5).
I thought that maybe I can do this:
$$ O_K \cong \mathbb{Z}[X] / (X^3 -2), $$ so
$$O_K / (7) \cong \mathbb{F}_7[X] / (X^3 -2), ~ O_K / (5) \cong \mathbb{F}_5[X] / (X^3 -2).$$
But I do not know what to do next. Can you help me?
Best Answer
I'm assuming you have to follow a somewhat elementary approach; otherwise the link by Sebastian Monnet to Dedekind's Factorization Theorem will help.
You're taking the right approach (and you're indeed going in the direction of that theorem). Those two rings $O_K/(7)$ and $O_K/(5)$, are they integral domains? (And hence even fields?) If they are, the ideals are prime; if they're not, then they have zero divisors. Lifting those zero divisors back to $O_K$ will tell you something about the factorization in $O_K$.