It is well known that if we take a random polynomial (monic with integer coefficients, say), then it will "almost always" be irreducible and have Galois group $S_n$ (see, for example, this MO question ). The tools in the proof are quite advanced however.
Consider the much weaker statement : (*) For every $n\geq 2$, there is an extension of $\mathbb Q$ with Galois group $S_n$.
(this statement is exactly what is needed to finish another recent MSE question )
So here goes my challenge : find a proof of (*) that's as elementary and self-contained as possible (but not simpler as Einstein would say).