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Are there any degree 7 polynomials over $\mathbb{Q}$ having Galois group $S_7$? If so, is there one for which this is easy to check with pencil and paper?

I know that for degree 3 polynomials, the Galois group will be $S_3$ if the discriminant is not a square in $\mathbb{Q}$, but I don't think the same holds for a degree 7.

Pascal's Wager
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    http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/galoisSnAn.pdf – Will Jagy May 24 '18 at 01:39
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    “Septic” sounds really wrong to me. – amd May 24 '18 at 02:16
  • @amd I think I used heptic in an early article. I think those are the only two choices; also, I'm not sure anyone else has ever gone with heptic, which also does not seem to be in the dictionary. Also resembles the adjective meaning related to the liver – Will Jagy May 24 '18 at 02:29
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    Why the only choices? I find "7th degree" works quite well. – C Monsour May 24 '18 at 03:14
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    @amd Septic definitely has an unfortunate double meaning. – Cheerful Parsnip May 24 '18 at 03:14
  • See this answer to another question for a proof that in fact, the odds of not having Galois group $S_7$ are asymptotically very low. –  May 24 '18 at 03:46
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    We say quartic not tetric; quintic not pentic. So Latin rules in this case, and septic it is, not heptic. – Lubin May 24 '18 at 03:48
  • As so often in the past, John, you've given me a smile, – John Hughes May 24 '18 at 04:35
  • in case anyone is interested, I worked out Gauss's procedure for producing cyclic septics, at least with "denominator" prime; my list at https://math.stackexchange.com/questions/1996552/any-more-cyclic-quintics/2033565#2033565 here is the chapter by David A. Cox on this, in modern language: http://zakuski.utsa.edu/~jagy/cox_galois_Gaussian_periods.pdf – Will Jagy May 24 '18 at 16:55

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Yes, there are even infinitely many examples with integer coefficients. This is a consequence of the Hilbert Irreducibility Theorem. See Serre's treatment of this at http://www.msc.uky.edu/sohum/ma561/notes/workspace/books/serre_galois_theory.pdf. There is also an article on the Hilbert Irreducibility Theorem in the latest number of the American Mathematical Monthly.

C Monsour
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Let $f$ be a polynomial with integer coefficients. It is a theorem that, if we reduce $f$ mod $p$ and $f$ contains an irreducible factor in $\mathbb{F}_p[x]$ of degree $k$, then the corresponding Galois group (when viewed as a subgroup of $S_n$) contains a $k$ cycle.

As a transposition and a $7$ cycle generate $S_7$, it suffices to find a polynomial of degree $7$ which, when reduced mod one prime, is irreducible, and when reduced mod another contains a factor of degree $2$. With some work, you can show $f(x)=x^7+x+1$ is irreducible mod $2$, but contains a quadratic irreducible factor mod $7$, so this gives a concrete example.

TomGrubb
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