if $K(a)=K(b)$ are $a$ and $b$ conjugates of the same minimal polynomial in $K[x]$?

Question

Let $K$ be a field and $F$ an extension of $K$, we know that if $a\in F$ and $b\in F$ are conjugate elements then with respect to some minimal polynomial in $K[x]$ then $K(a)=K(b)$. Is the reverse direction true as well? I.e if $K(a)=K(b)$ are $a$ and $b$ conjugates of the same minimal polynomial in $K[x]$?