This is a pretty basic question as I'm a beginner but I seem to be confusing myself. Suppose we have some $\alpha$ in an extension $K$ of $F$ such that $[F(\alpha):F]=2$. Am I safe to conclude that the minimal polynomial of $\alpha$ is of degree two?
To me, it makes sense as $[F(\alpha):F]=2$ being finite means that $\alpha$ is algebraic over $F$. But if its minimal polynomial was not degree two, but instead one, for example, then we would have $[F(\alpha):F]=1$, a contradiction.
I am a little hesitant because the textbook I've looked at mentions only the fact that
Hence $\alpha$ is algebraic over $F$ and satisfies a nonzero polynomial of degree at most $[F(\alpha):F]$
which seems to imply that it is possible for the degree of the minimal polynomial to be less than $[F(\alpha):F]$. Is that possible?