Proposition. the field of fractions of $F[S]$ is $F(S)$ [1302]
Proposition. the field of fractions of $F[S]$ is $F(S)$ [1302]
Given $K / F$ and $S \subseteq K$, let $$ F[S] = \{ \text{finite sums of }a \cdot \alpha_{1} \alpha_{2} \dots \alpha_{n} \mid a \in F, \alpha_{1}, \dots, \alpha_{n} \in S, n \in \mathbb{Z}_{\geq 0}\}. $$ Then $F[S]$ is a subring of $K$ and $$ F(S) = \left\{ \left. \frac{a}{b} \right\vert a, b \in F[S], b \neq 0 \right\}. $$ ($F(S)$ is the fraction field of $F[S]$, and it is exactly the same object as in generated fields.)
Taking multiples ensures $F[S]$ is closed under multiplication. Taking finite sums then ensures $F[S]$ is closed under addition. Moreover, we can take $a = 1$ and $n = 0$, then $F[S]$ contains $1$. It is easy to verify that $F[S]$ is actually an integral domain, thus we can take its field of fractions.