abstract algebra - Splitting field condition for all roots of irreducible polynomial

Let $F$ be a field, and $K$ a finite extension of $F$. Suppose that for every irreducible polynomial $P(x)\in F[x]$, if $P(x)$ has one root in $K$, then $P(x)$ has all its roots in $K$. How can we show that $K$ is a splitting field of some polynomial in $F$?

Since $K$ is a finite extension of $F$, there exists $c\in K$ such that $K=F(c)$. Let $P(x)$ be the minimal polynomial of $c$ over $F$. I think $K$ is a splitting field of $P$, but how to show it?

Answer

Nothing left to prove. $P(x)$ is irreducible over $F$, $\,K$ contains all its roots, and $K$ is the extension with (one of) them over $F$.