I know that finite extensions imply algebraic extensions (but not the converse) and that transcendental extensions are not necessarily finite extensions (even if they are simple extensions), so I am wondering if there is some mistake in my understanding that a simple algebraic extension is always a finite extension. Specifically, my claim is that:
If $\alpha$ is algebraic over F, then $[F(\alpha):F] < \aleph{}_0$ (finite degree)
My thoughts: Since $\alpha$ is algebraic, it has a unique minimal polynomial in $F$, whose existence implies that $[F(a):F]$ is finite since $F(a) \approx F[x]/\langle{}p(x)\rangle{}$.