Let $R$ be a ring, $K$ a subfield of $R$, and $x \in R$. Let $F(X)$ be the minimal polynomial of $x$ over $K$.
I want to prove that:
$K[x]$ is a field $ \iff K[x]$ is an integral domain $\iff F(X)$ is irreducible,
using the following lemmas:
If $B$ is an integral domain and $A$ a subring of $B$ such that $B$ is integral over $A$, then one has equivalence $A$ is a field $\iff B$ is a field.
$K[X]/(F(X))$ and $K[x]$ are isomorphic.
I need help especially with the last equivalence. How could the minimal polynomial not be irreducible?