If I have a subfield $F$ of a field $E$, and an algebraic (over $F$) $\alpha\in E$, I can form $F(\alpha)$ which is isomorphic to $F[x]/\langle f(x)\rangle$ for $f(x) = irr(\alpha, F)$. That is, $f(x)$ is the minimal degree and monic element of $F[x]$ such that $f(\alpha) = 0$.
The book I'm using defines $F(\alpha)$ officially as the image of $F[x]$ under $\phi_{\alpha}$, the map $f(x)\mapsto f(\alpha)$, and the isomorphism mentioned above comes from the fact that its kernel is $\langle f(x)\rangle$.
My question is: if there are other roots of the polynomial $f(x)$ in $E$, then are they necessarily contained in $F(\alpha)$? My intuitive guess is yes, since the field $F[x]/\langle f(x)\rangle$ doesn't know the difference between distinct roots of $f(x)$. But if this is correct then why?