Let $F$ be a subfield of $\mathbb{C}$ and $a,b \in \mathbb{C}$ be algebraic elements over $F$. Show that there exists an integer $n$ such that $a+nb$ is a primitive element of the field $K=F(a,b)$.
We know that the characteristic of the subfield is same as that of the field and hence $char\; F=0$ implies $F(a,b)$ is separable extension. Now, also $F(a,b)$ is a finite extension and hence $K/F$ has a primitive element (since it is finite and separable extension). So, the existence of a primitive element is guaranteed.
But how do I show the existence of integer $n$ such that $a+nb$ is a primitive element of the field $K=F(a,b)$. Can someone please help me ? Thankyou.