Construct, for every \(n\ge2\), a polynomial over \(\QQ\) whose Galois group is \(S_n\). Give an explicit example for \(n=7\).
Solution
A standard uniform family is \[ f_n(x)=x^n-x-1. \] A theorem of Nart–Vila, later proved independently by Osada, states that for every \(n\ge2\) the splitting field of \(f_n\) over \(\QQ\) has Galois group \[ \operatorname{Gal}(f_n/\QQ)\cong S_n. \] Selmer had earlier proved that \(x^n-x-1\) is irreducible for every \(n\ge2\).
Thus one may take \[ \boxed{f_n(x)=x^n-x-1.} \]
In particular, for \(n=7\), \[ \boxed{x^7-x-1} \] has Galois group \(S_7\) over \(\QQ\).
One route to the Galois-group theorem is: irreducibility makes the Galois group transitive; the ramification analysis shows that the group is generated by transpositions; and a transitive subgroup of \(S_n\) generated by transpositions is all of \(S_n\).