Transitivity, one transposition, and one long cycle often force a polynomial Galois group to be S_n. A small computation will anchor the general statement before the abstraction takes over.
Set-up
For a separable polynomial \(f\in K[x]\), the Galois group permutes its roots faithfully. Factorisations, discriminants, and resolvents constrain the resulting subgroup of \(S_n\).
The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).
The calculation
The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.
What survives abstraction
The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.
The boundary
A discriminant square distinguishes containment in \(A_n\), but it usually does not determine the entire Galois group by itself.
I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.