# Computing the degree of a finite morphism $\mathbb{P}^n\to \mathbb{P}^n$

Let $k$ be an algebraically closed field. Suppose that $f\colon \mathbb{P}^n(k)\to \mathbb{P}^n(k)$ is a morphism of the form $f = [f_0:\cdots: f_n]$ where the $f_i$ are homogeneous polynomials of degree $d$ with no nontrivial common zeros. In this case, the degree of the morphism $f$ is $d^n$. The only way I know how to compute this is via the intersection theory of $\mathbb{P}^n(k)$. Since the degree of $f$ is defined (very concretely) as the degree of the field extension $f^*k(\mathbb{P}^n)\subseteq k(\mathbb{P}^n)$, I wonder if there is a less high-tech way of computing that $\deg f = d^n$. Does anyone know a way?

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What about the Finiteness Theorem (if $k$ has characteristic $0$)? The pre-image of a generic point has size $d^n$ –  Jonathan May 8 '12 at 2:47