The Stacks project

Lemma 29.57.8. Let $f : X \to Y$ be an étale morphism of schemes. Let $n \geq 0$. The following are equivalent

1. the integer $n$ bounds the degrees of the fibres,

2. for every field $k$ and morphism $\mathop{\mathrm{Spec}}(k) \to Y$ the base change $X_ k = \mathop{\mathrm{Spec}}(k) \times _ Y X$ has at most $n$ points, and

3. for every $y \in Y$ and every separable algebraic closure $\kappa (y) \subset \kappa (y)^{sep}$ the scheme $X_{\kappa (y)^{sep}}$ has at most $n$ points.