Adjoining subfields

Question

Let $K$ be an extension of $F$ and $\alpha\in K$. Let $F(\alpha)$ be the smallest subfield in $K$ which contains $F$ and $\alpha$. Let $S:= \{q\in K: q= f(\alpha)/g(\alpha) f,g\in F[\alpha] ,g(\alpha)\neq0\}$ So my question is $F(\alpha)=S?$ I didn't understand the answer to the question named "Algebraic field extensions: Why $k(\alpha)=k[\alpha]?$ ".