Degree of Minimal Polynomial No Greater Than Degree of Extension

Proof

Clearly for $\alpha \in \mathbb{K}$, we have that

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 $\mathrm{deg}(f_{\alpha })=[\mathbb{F}(\alpha ):\mathbb{F}]\le [\mathbb{K}:\mathbb{F}]$.