I'm not really understanding the difference between $F[\alpha$] and $F(\alpha$), where $\alpha$ is an element of $E$, a field extension of $F$. From what I read, $F[\alpha$] = $F(\alpha$) when $\alpha$ is an algebraic over $F$. But why is that? And if $\alpha$ is transcendental, then it seems to be a completely different case.
Really, any clarification on what the difference between $F[\alpha$] and $F(\alpha$) is would be tremendously helpful. Thank you!