Proof de unique factorizaation in Dedekind Rings. Algebraic Number fields, Janusz, Second edition.
In the above proof, Theorem 3.13.
Why of the corolary 3.7, $\mathfrak{p}_1^{b_1}\cdots\mathfrak{p}_n^{b_n}\subseteq \mathcal{U}$ ?
I don´t understand this part of the theorem.
Thank you all.
I think the point is $\overline{\frak p}_1,\dots,\overline{\frak p}_n$ are all of the prime ideals in $R/\frak U$, and they correspond to prime ideals $\frak p_1, \dots, \frak p_n$ in $R$. Now by the corollary there are $b_1,\dots,b_n$ such that ${\overline{\frak p}_1}^{b_1}\cdots{\overline{\frak p}_n}^{b_n}$ equals the zero ideal in $R/\frak U$ (if some $b_i$'s are zero just multiply both sides by $\frak p_1, \dots, \frak p_n$ to make sure they are all positive). Now ${\overline{\frak p}_1}^{b_1}\cdots{\overline{\frak p}_n}^{b_n}=(0)$ in $R/\frak U$ $\Rightarrow$ ${\frak p}_1^{b_1}\cdots{\frak p}_n^{b_n}\subseteq \frak U$ in $R$.