In general an irreducible polynomial of degree $n$ over $\mathbb{Q}$ has Galois group $S_n$.
Could this be argued as follows:
A general polynomial $f(x)=a_0 + a_1 x+ \ldots + a_n x^n \in \mathbb{Q}[x]$ can be thought of as a polynomial with coefficients in $K=\mathbb{Q}(a_0,a_1,\ldots,a_n)$ where $a_i$'s are indeterminates. In the splitting field for this polynomial, say $E$, the polynomial factors as $f(x)=\prod_{i=1}^n (x-\lambda_i)$ and $a_i$'s are symmetric polynomials in $\lambda_i$. Now the Galois group of this extension $Gal(E/K)$ is $S_n$.
Is this the correct sense in which the above statement holds. If yes, then can it be made more precise. Specifically when I say "thought of" in the above statement.
If not, then what is the correct way to formalize the statement.
I do not know of any reference where this is specifically said. I just sort of believe it. Please let me know if I am wrong.
I suppose one way to say this would be the following.
For a generic polynomial $p(x)$ of degree $n$ in $\mathbb{Q}(a_0,a_1,\ldots,a_n)[x]$, there is a Zariski dense subset $A\subset \mathbb{Q}^{n+1}$ such that the specialization $f(a'_0,a_1',\ldots, a_n')$, where $a'=(a'_0,a_1',\ldots, a_n')\in A$, is irreducible in $\mathbb{Q}[x]$.
This is a consequence of Hilbert's Irreducibility Theorem.