Adjoining Elements: The field $F(S)$

Question

I am studying Galois Theory and I try to have a deeper knowledge. I am studying about the subfield which occurs if we adjoin a set in a field. Of course all textbooks have this notion, but none of them in a complete detail, so I tried to summarise the things I read in the following proposition.

Let $E/F$ be a field extension and let $S\subseteq E$. We write $\mathcal{F}_S$ to denote the set of all subfields of $E$ containing the field $F$ and the set $S$ (e.i. the set $F\cup S$). That is, $$\mathcal{F}_S:=\{K:\ K\leq E \text{ and } F\cup S \subseteq K\}.$$ Apparently $\mathcal{F}_S\neq \emptyset$, since $E\in \mathcal{F}_S$. We define the field\footnote{Obviously this is a field, as an intersection of fields.} $F(S) \leq E$ as the intersection of all the subfields of $E$ containing the field $F$ and the set $S$ (e.i. the set $F\cup S$). That is $$\boxed{F(S):=\bigcap_{K\in \mathcal{F}_S} K}\ .$$

Proposition. Let $E/F$ be a field extension and let $S\subseteq E$ be a subset of $E$.

1. The field $F(S)\leq E$ is the is the smallest subfield of $E$ containing the field $F$ and the set $S$.
2. The subset of all elements of $E$ that can be expressed as "quotients" of finite linear combinations with coefficients in $F$ of finite products of elements of $S$, \begin{align*} \mathfrak{X}\quad = \quad & \Big\{(a_0c_0+a_1c_1+\dotsb+a_nc_n)(b_0c_0+b_1c_1\dotsb+b_mc_m)^{-1} \in E:\ c_i\in S,\ a_i,b_j\in F, \\ & 0\leq i \leq n,\ 0 \leq j \leq m,\ n,m\in \mathbb Z^+ \text{ and } b_0c_0+b_1c_1\dotsb+b_mc_m \neq 0_E \Big\} \\ \quad = \quad & \Big\{f(c_0,c_1,\dots,c_n)g(c_0,c_1,\dots,c_m)^{-1}\in E:\ c_i\in S,\ f\in F[X_1,\dots,X_n],\ \\ & g\in F[X_1,\dots,X_m],\ n,m\in \mathbb Z^+ \text{ and } g(c_0,c_1,\dots,c_n)\neq 0_E \Big\} \end{align*} is a subfield of $E$, which contains $F$ and $S$. 3. It is $F(S)=\mathfrak{X}$.

Proof.

1. Suppose that $L\leq E$ is a subfield of $E$, such that $F\cup S\subseteq L$ \textit{or} equivalently $L\in \mathcal{F}_S$. Thus, since the subfield $L$ takes part in the intersection $\bigcap_{K\in \mathcal{F}_S}K$, we obtain $\bigcap_{K\in \mathcal{F}_S}K=:F(S)\subseteq L$. $\checkmark$
2. Denote by $P$ the set of all finite linear combinations of finite products of elements of $S$. If $p,q \in P$, then $p \pm q, pq \in P$. Hence, if $x := pq^{-1},y:=rs^{-1}\in \mathfrak{X}$, with $p, q, r, s \in P$ and $q, s \neq 0_E$, we see that $xy^{-1} = (ps-qr)(qs)^{-1} \in \mathfrak{X}$ and $xy^{-1}=(ps)(qr)^{-1} \in \mathfrak{X}$, where provided $y \neq 0_E$. Thus, $\mathfrak{X}\leq E$ and obviously $F,S\subseteq \mathfrak{X}$. $\checkmark$
3. From 1. and 2. we can deduce that $F(S)\subseteq \mathfrak{X}$. Also, $F(S)\subseteq \mathfrak{X}$, since any subfield containing $F$ and $S$ must contain all finite products of elements in $S$, all linear combinations of such products, and all ``quotients" of such linear combinations. In short, it must contain $\mathfrak{X}$. Therefore, $F(S)=\mathfrak{X}$. $\checkmark$

Questions.

1. Is everything above correct?

2. We define the smallest subring $A[S]$ containing the subring $A\subseteq R$ of $R$ and the set $S\subseteq R$ of $R$ and it will be $$A[S]=\{f(c_1,\dots,c_n)\in R: c_1,\dots,c_n \in S, n\in \mathbb Z^+\}.$$ Right?

3. In our first proposition, if we construct the subring $F[S]$, it is true that $\mathrm{Quot}(F[S])\cong F(S)$ via the isomorphism $$\varphi:F(S)\longrightarrow \mathrm{Quot}(F[S]),\quad x:=pq^{-1}\longmapsto \varphi(x)=\varphi(pq^{-1}):=\frac{p}{q}.$$ True?

Every further reading is very welcome.