Factorising a principal ideal where its norm is $p^k$ I was considering the following example:
Let $K=\mathbb{Q}(\sqrt{-29})$ and so $O_K=\mathbb{Z}[\sqrt{-29}].$ So by Dedekind's theorem, I wanted to decompose the principal ideal $(5)$ and I obtained $(5)=(5,\sqrt{-29}-1)(5,\sqrt{-29}+1):=P_1P_2 $ and that both prime ideals have norm $5$.
Then my book said
"Consider the ideal $(3+2\sqrt{-29})$, this has norm $125$ and so must be either $P_1^3, P_1^2P_2, P_1P_2^2$ or $P_2^3.$"
Why does this follow? I know that we can prime decompose this ideal and that $P_1$ and $P_2$ are the only ideals with norm $5$. But why did we rule out the possibility that $(3+2\sqrt{-29})$ is already prime and the possibility that $(3+2\sqrt{-29})=QP_1$ where $Q$ is a prime ideal with norm $25$, say?
I am thinking I must have missed some small details, many thanks in advance for lending a hand!
 A: In a quadratic field there can be at most two prime ideals above any rational prime see: https://en.wikipedia.org/wiki/Splitting_of_prime_ideals_in_Galois_extensions#The_Galois_situation. Using the notation there $\text{degree} = e \cdot f \cdot g$ we have the degree is $2 = 1\cdot1\cdot2$ already, having any more primes would contradict this, so the fact you found two distinct  primes above 5 means these are the only primes above 5.
Even if your field wasn't Galois some similar arguments can be made and we always have that the degree of the field is always greater or equal to the number of primes above any prime in the base field.
I don't know if this has already been introduced in your book though :) which book is it?
A: Note that the norm of an ideal is defined by the size of $O_K/I$ for $I$ an ideal. Thus by Lagrange's Theorem, we must have $N(I).x\in I$ for each $x\in O_K$, where $N(I)$ is the norm of the ideal $I.$
Now, this implies that $N(I)\in I$ for each ideal $I.$ Thus we have $I|(N(I)).$
So apply to this case, we have $(3+2\sqrt{-29})|(125)=(5)(5)(5)=P_1^3P_2^3.$ Now, by considering the norm on the ideal $(3+2\sqrt{-29})$, we deduce it must have the required form.
