Question about a proof in Serre’s "Local fields" of unique factorization of a fractional ideal of a Dedekind Domain I'm trying to understand Serre's proof (in Local Fields) of the fact that a fractional ideal $\mathfrak{a}$ of a Dedekind domain factors uniquely as $\mathfrak{a} = \prod_{\mathfrak{p}}\mathfrak{p}^{v_{p}(\mathfrak{a})}$.
The approach he took was that since $v_{\mathfrak{p}}(\mathfrak{a}) = 0$ for all but finitely many $\mathfrak{p}$, say, $v_{\mathfrak{p_{j}}}(\mathfrak{a}) = e_{j} \neq 0$ for $j = 1,\ldots, k$. Let $\mathfrak{b} = \prod_{j=1}^{k}\mathfrak{p_{j}}^{e_{j}}$. Then show $\mathfrak{a} = \mathfrak{b}$ locally, i.e, $\mathfrak{a}A_{\mathfrak{p}} = \mathfrak{b}A_{\mathfrak{p}}$ for all prime $\mathfrak{p}$; hence, they must be the same. I have been able to fill in most details except for the fact that  $\mathfrak{a} = \mathfrak{b}$ locally. If they are integral ideal of $A$, I can prove it, but since they are fractional with possibly negative powers, I could not figure out how to prove that claim. Note that Serre seems to use the following definition for $v_{\mathfrak{p}}(\mathfrak{a})$ for a fractional ideal $\mathfrak{a}$:https://mathoverflow.net/questions/52198/what-is-the-definition-of-the-valuation-of-a-fractional-ideal and I have trouble applying that definition here. Any help is greatly appreciated. 

The local ring $A_\mathfrak{p}$ is a DVR and in particular a principal ideal domain. So the fractional ideal $A_{\mathfrak{p}}\mathfrak{a}$ is a free  $A_{\mathfrak{p}}$-module (since finitely generated and torsion free) of rank 1,
say with basis element $x$ . Any other basis element $x' \in A_{\mathfrak{p}}\mathfrak{a}$ must have the form $x' = ux$ for a unit $u \in A_{\mathfrak{p}}$.
Now, the valuation $v_{\mathfrak{p}}$ is a function $K^\times \to \mathbf{Z}$
where $K$ is the field of fractions of $A$ and also of $A_{\mathfrak{p}}$. 
So $v_{\mathfrak{p}}(\mathfrak{a}) := v_{\mathfrak{p}}(x)$ is well-defined.
And I believe that you will find that this definition of $v_{\mathfrak{p}}(\mathfrak{a})$ should make clear what is required for your question.
