The Stacks project

Definition 42.41.1. Let $k$ be a field (Example 42.7.2). Let $p : X \to \mathop{\mathrm{Spec}}(k)$ be proper. The degree of a zero cycle on $X$ is given by proper pushforward

(Lemma 42.20.3) combined with the natural isomorphism $\mathop{\mathrm{CH}}\nolimits _0(\mathop{\mathrm{Spec}}(k)) = \mathbf{Z}$ which maps $[\mathop{\mathrm{Spec}}(k)]$ to $1$. Notation: $\deg (\alpha )$.