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I'm studying the proof that for every $n\in \mathbb{N}$, there exists a polynomial $f\in \mathbb{Q}$ such that $\mbox{Gal}(E/\mathbb{Q})\cong S_n$, with $E$ the splitting field of $f$ over $\mathbb{Q}$. I have to make a presentation about this so I was interested to show some example for $n$ not a prime greater than 5. ($S_6$ for example). My problem is that the proof doesn't show how to find such polynomial.

Can you help me to construct it or directly provide me with some polynomial that do this?

jiyanez
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  • I gave an example in this answer. Unfortunately it depends on the piece of theory explained by Qiaochu Yuan here, so I'm not sure how useful it is to you. That bit is not covered in all courses on Galois theory. If you can explain the result early in your presentation that might be ok? – Jyrki Lahtonen Dec 02 '14 at 21:58
  • But @JyrkiLahtonen, isn’t it the case that almost any sixth-degree polynomial should give $S_6$ for the Galois group? Like, maybe $X^6-X-a$ for almost any integer $a$, like, maybe, $a=1$? Of course I would quail at the prospect of verifying this for any specific polynomial. – Lubin Dec 02 '14 at 22:06
  • I suspect that you're right @Lubin. And I share your excitement about having to prove any single example. I crafted that example polynomial carefully so that it factors in a desired way modulo selected small primes. Precisely so that I could apply Dedekind's theorem. Probably there is a simpler way of producing an example, but I don't know one. – Jyrki Lahtonen Dec 02 '14 at 22:09

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