Intuitively, it seems that given a random irreducible $p(x) \in \mathbb{Q}[x]$ of degree $n$, the Galois group of $p(x)$ over $\mathbb{Q}$ should be $S_n$. Otherwise, if $\alpha_1,...,\alpha_n$ are the roots of $p(x)$, there will exist $k < n-1$ such that $q_k(x) := \dfrac{p(x)}{(x-\alpha_1)...(x-\alpha_k)}$ factors over $L_k := \mathbb{Q}[\alpha_1,...\alpha_k]$. This implies some nontrivial algebraic dependency between the roots, beyond them all satisfying the same irreducible polynomial. Intuition from linear algebra (and other fields of math) suggests that imposing a nontrivial algebraic relation tends to reduce the dimension, or size, of the possible solution space, so that the Galois group should be the full symmetric group with very high "probability".
I am not sure of the best way to formalize this intuition. One possible route would be to impose some kind of height function on polynomials in $\mathbb{Q}[x]$, so that there are only finitely many polynomials with height less than a given constant, and look at the probability of an irreducible polynomial having Galois group $S_n$ as the height goes to infinity. But this seems a little crude somehow. One could also try to show that the set of algebraic numbers with Galois group the full symmetric group is somehow "open" and "dense", or that the property of having symmetric Galois group is stable under perturbation. Maybe there is a topology (or related structure) on $\overline{\mathbb{Q}}$ which is amenable to this type of reasoning. Is anyone familiar with any results in this vein? I am also interested in partial results, or comparable situations working over other base fields. Thanks.
EDIT: I see there are substantial results on polynomials with integer coefficients, typically using a height function. Can we say anything about polynomials with rational, non-integer coefficients?
