Lemma 29.51.9. Let $X$, $Y$, $Z$ be integral schemes. Let $f : X \to Y$ and $g : Y \to Z$ be dominant morphisms locally of finite type. Assume that $[R(X) : R(Y)] < \infty $ and $[R(Y) : R(Z)] < \infty $. Then

\[ \deg (X/Z) = \deg (X/Y) \deg (Y/Z). \]

**Proof.**
This comes from the multiplicativity of degrees in towers of finite extensions of fields, see Fields, Lemma 9.7.7.
$\square$

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