an easy algebra question

Question

$\mathbb{K}$ is a field extension of field $\mathbb{F}$ and $a\in \mathbb{K}$ and if $p(x) = min (\mathbb{F},a)$, why $\mathbb{F}[x]/(p(x))$ is isomorphic to $\mathbb{F}[a]$?

Answer 1

Consider $\phi:\mathbb{F}[X]\to \mathbb{K}$ defined by $\phi(P)=P(a)$. This ring homomorphism has by definition of $\mathbb{F}[a]$ as range. Its kernel is by definition the ideal generated by the minimal polynomial. Whence the isomorphism.