Let $X$ be a curve defined over a number field $K$ and let $\pi: Y \rightarrow X$ be a finite etale cover. For $x \in X(K)$ put $E_x$ for the ring of regular functions of the fibre $\pi^{-1}(x)$. Then there is a claim (see page 29 of this paper, in the displayed expression above (6.1)) that $E_x$ (an etale $K$-algebra) decomposes into a product of fields
$$\displaystyle E_x = \prod_{x' \in \pi^{-1}(x)} K(x').$$
Here the meaning of $K(x')$ is not explained in the paper as far as I can see, but from context I believe it is the field $K'$ obtained by adjoining all of the coordinates of $x'$ to $K$. Note that $\pi^{-1}(x)$ is a zero-dimensional scheme, so that it consists of finitely many points defined over a finite extension of $K$.
Can anyone explain this decomposition?
First of all, in that paper $E_x$ is the ring of regular functions on the fibre $\pi^{-1}(x)$, which is a zero dimensional $K$-scheme, even reduced. Such scheme is a union of finitely many closed points $x$, so, as schemes, of the form $x\cong\operatorname{Spec}(K')$, for $K(x):=K'$ a finite extension of $K$. The paper is only saying what I said, i.e. $$E_x\cong\operatorname{Spec}(\prod_{x' \in \pi^{-1}(x)} K(x'))$$ as $K$-scheme.