a basis for the simple extension $F(\alpha)$ over $F$

Question

Let $E$ be a field extension of $F$. Suppose $\alpha\in E$ is transcendental over $F$. Is it true that $\{\alpha^n:n\in\Bbb Z \}$ is a basis for the simple extension $F(\alpha)$ over $F$? In fact, I want to write a basis explicitly for $F(\alpha)$ over $F$. Thanks!

Answer 1

No, it is not true. How would you express, say $\frac1{1+\alpha}$ as a linear combination of elements of the form $\alpha^n$ with $n\in\mathbb Z$? If you could do it, then $\alpha$ would be algebraic.