I want to understand the proof that the number fields (which are dedekind domains) have unique factorization of ideals. I am trying to read this proof here IDEAL FACTORIZATION - KEITH CONRAD but..
I couldn't understand the reason for the inclusions in this proof:
Corollary 3.5: For any nonzero ideal $\mathfrak b$ and nonzero prime ideal $\mathfrak p$, $\mathfrak {pb} \subseteq \mathfrak b$ with strict inclusion.
Proof. Easily $\mathfrak {pb} \subseteq \mathfrak b$. If $\mathfrak {pb} = \mathfrak b$ then for all $k\ge1$, $\mathfrak b = \mathfrak p^k \mathfrak b$ therefore $\mathfrak b \subseteq \mathfrak p^k$. rest of proof omitted
First of all these are all ideals of $\mathbb O_K$ the ring of integers of a number field $K$.
How do we know $\mathfrak {pb} \subseteq \mathfrak b$?
How do we know $\mathfrak {b} \subseteq \mathfrak p^k$?
Thank you very much.