My question refers to an example of the normalisation process of a scheme as introduced at wiki: https://en.wikipedia.org/wiki/Normal_scheme
The statement is that $X'={\displaystyle {\text{Spec}}(\mathbb {C} [x,y]/(x)\times \mathbb {C} [x,y]/(y))}$ is the normalisation of $X := Spec(\mathbb {C} [x,y]/(xy))$. Why?
To show this it suffice to see that locally (here: at each localisation of a prime ideal $\mathcal{p} \subset \mathbb {C} [x,y]/(xy)$ the ring
$$(\mathbb {C} [x,y]/(x)\times \mathbb {C} [x,y]/(y)))_{\mathcal{p}} \cong (\mathbb {C} [x,y]/(x))_{\mathcal{p}} \times (\mathbb {C} [x,y]/(y)))_{\mathcal{p}}$$
is exactly the normalisation of $$(\mathbb {C} [x,y]/(xy))_{\mathcal{p}}$$
therefore $(\mathbb {C} [x,y]/(x)\times \mathbb {C} [x,y]/(y)))_{\mathcal{p}} $ is the integral closure of $(\mathbb {C} [x,y]/(xy))_{\mathcal{p}}$ the total ring of fractions $Q((\mathbb {C} [x,y]/(xy))_{\mathcal{p}})$.
I don't why does it hold.