# Show as simply as possible that $\mathbb Q$ has an extension with Galois group $S_n$

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).

• It's not so hard for $n$ prime. All you need is an irreducible polynomial of degree $n$ with exactly two nonreal roots. – Gerry Myerson Oct 8 '15 at 11:48
• I found another exposition, pp 104-105 of Richard Koch's notes on Galois Theory, available at pages.uoregon.edu/koch/Galois.pdf – it's still longer than anything I'd want to type out here, but skimming it (I haven't read it closely) it looks promising. – Gerry Myerson Oct 11 '15 at 8:07
• Makoto Kato wrote up an exposition at math.stackexchange.com/questions/165675/… – the last part of it looks very much like the last part of Koch's write-up. – Gerry Myerson Oct 11 '15 at 8:15
• – Gerry Myerson Oct 11 '15 at 8:21
• @GerryMyerson Thanks for all the links, I'll look them up when I have the time – Ewan Delanoy Oct 11 '15 at 8:37

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