Let F be a subfield of E, $\displaystyle \alpha\in E$and $\displaystyle \alpha$is an algebraic over $\displaystyle F$. Let $\displaystyle \phi_\alpha$be the evaluation homomorphism. $\displaystyle F(\alpha)$is the minimal field containing $\displaystyle F,\alpha$

True or false: $\displaystyle \phi_\alpha(F[x])\cong F(\alpha)$?

One main problem is:$\displaystyle \phi_\alpha(F[x])$is a field????