Let $F$ be a finite field extension of $K$. Then there exists a field $L$ containing $F$ and a polynomial $g(x)\in K[x]$ such that $L$ is a splitting field for $g(x)$ over $K$ and every irreducible factor of $g(x)$ in $K[x]$ has a root in $F$.
Let $\{v_1,\dots,v_n\}$ be a basis of $F$ over $K$. Let $g(x)$ be the product of the minimal polynomials of $v_1,\dots,v_n$ over $K$. Let $L$ be a splitting field of $g(x)$ over $F$. Then $L=F(r_1,\dots,r_m)$, where $r_1,\dots,r_m$ are the roots of $g(x)$.
Since $v_1,\dots,v_n\in \{r_1,\dots,r_m\}$ and $F=K(v_1,\dots,v_n)$, $L$ is a splitting field of $g(x)$ over $K$.
Question:
Are there any example such that $F$ is a proper subfield of the splitting field $L$?