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I'm trying to prove that one can always construct an nth degree polynomial with Galois Group $S_n.$ I've proven that one can always construct an nth degree polynomial with Galois Group Sn over the field $F_0(s_1,...,s_n)$ where $s_1,...,s_n$ are the elementary symmetric polynomials. I've also seen conditions which a polynomial must have to have Galois Group $S_n,$ but this doesn't prove existence. Does anyone have a proof that there is at least one $n^{th}$ degree polynomial with Galois Group $S_n?$

Thanks

EDIT: I am considering only polynomials over Q (rationals), not other fields in general.

Pianoman1234
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1 Answers1

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Hilbert showed that all symmetric and alternating groups can be realized as Galois groups of polynomials with coefficients in $\mathbb{Q}$.

A modern exposition of his argument is given in the Wikipedia page on the Inverse Galois problem, see the section Symmetric and Alternating Groups.

Francesco Polizzi
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  • This is interesting, but I would like to prove it without using already known theorems and lemnas which I can't prove myself. I would rather have a proof 'from scratch' if you see what I am getting at – Pianoman1234 Aug 21 '17 at 18:17
  • Could you please explain how on the wikipedia page they are able to claim that Gal(f(x, s)/Q(s)) is doubly transitive? – Pianoman1234 Aug 23 '17 at 09:10