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I'm just becoming interested in the use of programming in Mathematics (specifically group theory/number theory). Now, I was wondering whether there were any particular scripts (raw scripts) that were available in determining the Galois Group of certain polynomials. In fact, I was wondering whether there were particular scripts that could determine the suitable polynomials over Q associated with M11 (the simplest Mathieu group).

I am not looking for particular packages (e.g GAP/Sage), but rather the raw script for these packages such that if you were to put in M11 you could output an associated polynomial (whether over Q or as a polynomial in integers) and the specific method involved. The reason for this question is I would like to see the number of specific scripts that are available for determining M11 as a Galois group and the computational efficiency with respect to the different scripts. Eventually I would like to apply the scripts further to other finite groups and modify them but to see the skeleton for M11 would be tremendously helpful.

Thank you for your help!

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  • I'm not an expert on this, but I think this may be somewhat challenging. Basically because the driving problem of Inverse Galois Theory, given a finite group $G$ construct a field $K$ such that $Gal(K/\Bbb{Q})\simeq G$, is still open in general! Wikipedia claims that no solutions are known for the Mathieu group $M_{23}$. Sounds incredible given that the Monster has been handled, but I am unable verify/refute that for lack of expertise. – Jyrki Lahtonen Jul 13 '17 at 06:45
  • When viewing the Galois group $G$ of a polynomial $f(x)$ of degree $n$ as a subgroup of $S_n$ it is easy to force permutations of a desired cycle structure to appear in the Galois group. All you need to do is to pick irreducible polynomials modulo various primes $p$ of degrees matching with the cycle lengths, and force $f(x)$ to have those factors modulo $p$. See here for a trivial example. The catch is to avoid permutations not in $G$. I don't know much about that. – Jyrki Lahtonen Jul 13 '17 at 06:54
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    Effective versions of Chebotarev's density theorem tells us, in principle, that if we can avoid certain types of factorizations up to high enough primes, then the corresponding permutations won't show. But I suspect that the bound on the size of primes depends on the size of the coefficients of the polynomial (or rather the discriminant of the splitting field, or ..?), and we would be trying to hit a moving target if mindlessly CRT-combining known factorizations modulo primes. Also, for non-abelian $G$ it is not clear what kind of connections between splitting patterns of various primes exist. – Jyrki Lahtonen Jul 13 '17 at 06:59
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    Such questions run very deep into non-abelian extensions of class field theory. At least that is the impression I have. Hopefully an expert shows up and can shed some light on this as well as point you to relevant literature. – Jyrki Lahtonen Jul 13 '17 at 07:01

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