My question is related to this one: Computing the Galois group of a polynomial.
I was wondering if there is a faster algorithm just to compute the order of the group rather than the group itself.
Also, has anybody compared the performance of GAP and Magma in computing Galois groups? I just heard Magma is very good at it.
I asked this question because I encounter every so often new bug with Magma's implementation and I wanted to see if I can implement something similar. But at this time I'm just interested in the exponent at the first place. This is the last annoying error that I get for basically any deg 5 poly that has Gal group $S_5$.
k := FiniteField(2); kx<x> := RationalFunctionField(k); kxbyb<y> := PolynomialRing(kx); MinP := y^5 + y + x^2 + x; print GaloisGroup(MinP);
The result is:
Runtime error: too much looping
Which I don't understand what it means (Magma Ver 2.16-8).
To be more clear, my ultimate goal is to check a lot of polynomials and throw out those with $S_n$ Gal group and focus on those which are not such. As you see even an upper bound over the exponent is enough for me.