Let $F$ a field and suppose that $E$ is a extension field of $F$. Now, take $\alpha\in E$ trascendental over $F$. My question is about the form of the elements of $F(\alpha)$. I think that $$F(\alpha)=\left\{\frac{f(\alpha)}{g(\alpha)}:f,g\in F[x]\right\}$$.
Am I right?
Yes, you are correct.
$F(\alpha)$ is defined to be the field of rational polynomials in $\alpha$ with coefficients in $F$.
You can also think about $F(\alpha)$ as being isomorphic to $F(t)$, the field of rational functions (the evaluation mapping at $\alpha$ serves as this isomorphism between transcendental elements and indeterminates), and $F(t)$ clearly has this form.