Degree of Minimal Polynomial No Greater Than Degree of Extension
Theorem
Let \(\mathbb{K}/\mathbb{F}\) be a finite field extension. Then for any \(\alpha \in \mathbb{K}\) with minimal polynomial \(f_\alpha\)
\[ \deg(f_\alpha) \leq [\mathbb{K} : \mathbb{F}].\]
Proof
Clearly for \(\alpha \in \mathbb{K}\), we have that
\[ \mathbb{F} \subseteq \mathbb{F}(\alpha) \subseteq \mathbb{K}.\]
Then from the multiplicativity of degree of field extensions and the fact that the degree of a simple extension is the degree of the minimal polynomial of the generator, we know that \(\deg(f_\alpha) = [\mathbb{F}(\alpha) : \mathbb{F}] \leq [\mathbb{K} : \mathbb{F}]\).