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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).

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Theorem 37 of Hadlock, Field Theory and Its Classical Problems, #19 in the MAA Carus Mathematical Monographs series, is

For every positive integer $n$, there exists a polynomial over $\bf Q$ whose Galois group is $S_n$.

The proof is only half a page long, BUT it comes as the culmination of a long chapter, which suggests to me that any self-contained proof will be too long for anyone to want to write it out here. But maybe you could have a look at it, to see whether you'd be able to post a summary (or to get a better idea of just how hard the question is).

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