Task: let $K$ be an extension of field $F$. Show, that $K$ include simple extension of field $F$.
I denote $K^{'}$ as simple field extension of field $F$. Definition of simple field extension: if $\exists c \in K^{'}\setminus F$ that $K^{'} = F(c)$.
I think that simple extension of field should be the smallest, so I rewrote it as $K^{'} = \cap \{K^{'}<K: c \in K^{'}, F \subset K^{'}\}$, so $K^{'} \subseteq K$, where $[K^{'}:F]$ - is random.
Is it correct?