Use Dedekind’s Theorem to factorise the following principal ideals in the ring of integers of the following fields.
a) $Q(√3): ⟨2⟩,⟨3⟩,⟨5⟩,⟨30⟩$
b) $Q( ^3√2): ⟨7⟩, ⟨29⟩, ⟨31⟩$
Here is what I understand about the theorem Let $[K:Q]=n$ with the ring integers $O_k=Z[ø]$
Given the minimal polynomial of $√3$, we have $x^2-3$. And for $^3√2$, we have $x^3-2$.
a) $p=2$; $x^2-3=x^2mod2$. By Dedekind, $⟨2⟩ = ⟨2, √3⟩$
$p=3$; $x^2-3=x^2-1mod3$. Of course, we know $x^2-1=(x+1)(x-1)$.
Hence $⟨3⟩ = ⟨3, 1+√3⟩•⟨3, 1-√3⟩$
$p=5$; $x^2-3=x^3-2+10mod5=(x+2)(x^2-2x+4)mod5$
$⟨5⟩ = ⟨5, 1+i⟩•⟨5, i^2-2i+4⟩$
The first three numbers generated were prime numbers. However, 30 is not. I'm thinking of factoring $30$ to $2, 3$, and $5$, I could use three of those prime numbers and it would give me a set of generated numbers by Dedekind. I am having trouble with applying Dedekind on part b). They're a little trickier than part a for I'm dealing with the cube root of 3.