Why does the definition of a simple extension make use of polynomial fractions?

Question

If we have a field $E$ which extends $F$, and $\alpha \in E$, then my textbook says we define the smallest field which contains $\alpha$ and $F$ to be $\left\{\frac{g(\alpha)}{h(\alpha)} \;:\; g, h \in F[x]\right\}$. Virtually all other definition I've seen online also do it this way. I was wondering why it's necessary to include the fraction in the definition, since for any $g, h \in F[x]$, we would have $\frac{g}{h} \in F[x]$ (assuming $h \neq 0$), because $F[x]$ is a field; so would it not be sufficient to just have the definition be $\left\{f(\alpha) \; : \; f \in F \right\}$?

Answer 1

No, you would not have $g/h$ in $F[X]$ as it is not a field, it is the ring of polynomials, which is not a field. The inverse of a non-constant polynomial is not a polynomial. The field of rational functions is usually denoted $F(X)$.

You could however write $\{k(a) \colon k \in F(X)\}$ instead.

Note that for certain elements $a$ (called algebraic over $F$) it will in fact be sufficient to consider polynomials only, yet this is a bit orthogonal to the question at hand.