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
k[x]/(P) \to k(\alpha ), \quad x \mapsto \alpha
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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