Dear GAP Forum,
I'm looking for a quick way to get the size of Galois group of an irreducible
integer polynomial f (to speed up PARI's function nfsplitting).
The method I'm currently trying is what we call "local method" comparing the
information of the factorization of f modulo prime numbers with the data of the
cyclic index polynomials generated from TransGrp. Here is my PARI/GP's code:
cnt(M) = [[ss, #select(tt -> ss == tt, M)]|ss <- Set(M)];
fast_nfsplitting(f) =
{
my(d = poldegree(f));
if(d <= 37 && (d-24)*(d-30)*(d-32)*(d-36) <> 0 && polisirreducible(f),
my(dc = poldisc(f), P = 0, L = List([]), fd, ci, ds, ans, dd,
CId = CI[d]); /* CI is a list s.t. [[[ 1/2*x_1^2+1/2*x_2, 1 ]],
[[ 1/3*x_1^3+2/3*x_3, 1 ],[ 1/6*x_1^3+1/2*x_1*x_2+1/3*x_3, 1 ]], ...] */
forprime(p = 2, 10^3,
if(Mod(dc, p) <> 0,
listput(L, fd = factormod(f, p, 1)[, 1]~);
if(#fd == d, P++;if(P == 3, break))));
ci = vecsum([vecprod([eval(concat("x_", j))|j <- s[1]])*s[2]|s <-
cnt(L)])/#L;
ds = vecmin([norml2(s[1]-ci)|s <- CId],&j);ans = CId[j];
print([ds*1., dd = 1/polcoef(ans[1], d, x_1), ans[2]]);
nfsplitting(f, dd),
nfsplitting(f))
};
/* for(i = 2, 20, fast_nfsplitting(x^i-2));
time = 960 ms.*/
However, in the case of DegreeAction = 24, 30, 32, 36,
it takes a considerable amount of time to generate the CycleIndex data files by
my GAP's code:
tst := function(G)
local s;
if IsSolvable(G) then s:=1; else s:=0; fi;
return([CycleIndex(G),s]);
end;;
for n in [2..37] do
fn := Concatenation("./CI.",String(n));
PrintTo(fn, "[");
L := AllTransitiveGroups(DegreeAction, n);
for G in L do AppendTo(fn, tst(G), ","); od;
AppendTo(fn, "]");
# After this, reshape using sed etc.
od;;
and even if all the data is obtained, it will be huge so it will take time to
search.
Is there any other better way to get the size of Galois group of f?
macsyma
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