Thank you for the reponse. I agree with what you said, but indeed the point that is blocking me is precisely how to determine the degree of the elements of this fractional ideal. I know that this degree is going to be N$(x)$/N$(\mathfrak{a})$ for every $x\in \mathfrak{a}$ ($\mathfrak{a}$ being the Hom space.) but since I don't explicitely know $\mathfrak{a}$, I don't know how to calculate the degree of its elemtents.