5

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).

Community
  • 1
Ewan Delanoy
  • 63,436
  • 1
    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 08 '15 at 11:48
  • I found another exposition, pp 104-105 of Richard Koch's notes on Galois Theory, available at http://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 08:07
  • Makoto Kato wrote up an exposition at http://math.stackexchange.com/questions/165675/constructing-a-galois-extension-field-with-galois-group-s-n – the last part of it looks very much like the last part of Koch's write-up. – Gerry Myerson Oct 11 '15 at 08:15
  • See also http://www.nptel.ac.in/courses/111101001/downloads/Lecture26.pdf also http://people.maths.ox.ac.uk/~vonk/2012JNT1.pdf – Gerry Myerson Oct 11 '15 at 08:21
  • @GerryMyerson Thanks for all the links, I'll look them up when I have the time – Ewan Delanoy Oct 11 '15 at 08:37

1 Answers1

1

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).

Gerry Myerson
  • 185,413