Doubt about Jacobson Basic Algebra I

Question

I'm sorry if this is a question entirely about a book: Basic Algebra I written by Jacobson (precisely I'm referring to the second edition).

In chapter 4.9 it is stated the famous Ruffini-Abel theorem: the general equation of the $n$-th degree is not solvable by radicals if $n>4$.

Now I quote the beginning of chapter 4.10:

The Ruffini-Abel theorem states that the general equation of degree $n \ge 5$ are not solvable by radicals. […] In spite of this result it is conceivable that all equations with coefficient in a field $F$ are solvable by radicals. In some case this is true. For example, it is trivially so if $F = \mathbb{R}$.

We shall now show that if $F = \mathbb{Q}$ and $p$ is any prime then there exist $f(x) \in \mathbb{Q}[x]$ having $S_p$ as Galois group. For $p\ge5$ these are not solvable by radicals.

My questions are:

Isn't this "In spite of this result it is conceivable that all equations with coefficient in a field $F$ are solvable by radicals. In some case this is true." contraddicting Ruffini-Abel's theorem?

How is trivial if $F = \mathbb{R}$?

Isn't this "We shall now show that if $F = \mathbb{Q}$ and $p$ is any prime then there exist $f(x) \in \mathbb{Q}[x]$ having $S_p$ as Galois group. For $p\ge5$ these are not solvable by radicals." a weaker statement of theorem 4.15: the Galois group of $f(x) =0 $ (the general equation of $n$th degree) is the symmetric group $S_n$ for every $n$?

Answer 1

Jacobson uses the following definition of solvability by radicals (Definition 4.2 in Chapter 4.7 “Galois' criterion for solvability by radicals”):

Definition. Let $f\in F[x]$ be monic of positive degree. Then the equation $f(x) =0$ is said to be solvable by radicals over $F$ if there exists an extension field $K/F$ which possesses a tower of subfields $$ F =F_1 \subset F_2\subset\cdots\subset F_{r+1} = K $$ where each $F_{i+1}=F_i(d_i)$ and $d_i^{n_i} = a_i\in F_i$, and $K$ sontains a splitting field over $F$ of $f$. A tower of subfields as above will be called a root tower over $F$ for $K$.

With this definition, it's indeed trivial for $F=\mathbb R$, because it's enough to set $F_2=K=\mathbb C$.

As for Theorem 4.15, it concerns the equation $f(x)=0$ as considered over the field $F(t_1,\dots,t_n)$ of rational functions over $F$ (where the variables are the coefficients of $f$), so its Galois group is computed over $F(t_1,\dots,t_n)$, whereas the statement of Theorem 4.16 concerns the existence of an individual equation over $\mathbb Q$ with the Galois group equal to $S_n$.