Lemma 9.9.2. The degree of the minimal polynomial is $[k(\alpha ) : k]$.

**Proof.**
This is just a restatement of the argument in Lemma 9.6.8: the observation is that if $P$ is the minimal polynomial of $\alpha $, then the map

is an isomorphism as in the aforementioned proof, and we have counted the degree of such an extension (see Example 9.7.6). $\square$

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