[Math] Field Extensions as $F$ adjoin some element

Question

Let $F$ be a field and $E$ an extension of $F$. Is it always possible to write $E=F(\alpha_1,\alpha_2,\ldots)$?

If $E$ is a finite extension then I think it is possible to write $E=F(\alpha_1,\alpha_2,\ldots,\alpha_n)$. My reason is that if we take $\alpha\in E$ then as $[E:F]<\infty$ for some $n$ we must have $\alpha^n\in\text{Span}\{\alpha, \ldots,\alpha^{n-1}\}$. Meaning that $\alpha$ satisfies an (irreducible) polynomial in $F[x]$. If we keep doing this for each element in $E$ then we get $E=F(\alpha_1,\alpha_2,\ldots,\alpha_n)$. Is this correct?

What about the case when $E$ is not a finite extension?

Thanks

Answer 1

An example is $\mathbb R$ as extension of $\mathbb Q$.