Are there some easily testable conditions that allow us to quickly deduce that a polynomial $f(X)\in\mathbb{Z}[X]$ of degree $n$ has Galois group $S_n$? Something that works in ''most'' cases? This seems to crop up a lot in computations I've been doing lately. I usually just reduce modulo a few primes to get some kind of cycles that must be in the Galois group and then try to look at degrees, but I'm wondering if there are some nice tricks that would help me avoid doing this?

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    $\begingroup$ For prime $n$ prime, see lhf's answer here; see also Keith Conrad's note here. $\endgroup$ – Arturo Magidin Nov 5 '11 at 22:18
  • $\begingroup$ Thanks, the result quoted by lhf I knew. Keith Conrad's article seems interesting. I'll take a look at it. $\endgroup$ – pki Nov 5 '11 at 23:01

Dummit and Foote has a section on computation of Galois groups over the rationals (chapter 14, section 8). Besides reducing modulo primes, they suggest analyzing the resolvent of your polynomial. The problem is that if your polynomial has large degree, the resolvent polynomial might be huge. You also need to know the transitive subgroups of $S_{n}$.

They cite http://www.maths.qmul.ac.uk/~leonard/mcompsci_soicher.pdf and it contains several examples.


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