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.
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.
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.
