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?