Hi Marta,
1) most systems, like Maple, can compute Galois groups, in some
standard format (a set of generating permutations, e.g.)
2) like Dima wrote, it's hopeless to compute for degree above a hundred;
and usually the limit is much lower
3) the expressions you gave don't produce irreducible polynomials. most
computer command require an irreducible polynomial.
Here's my sample code, in Maple:
A:=(1/sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x - sqrt(x^2-4*x))/2)):
B:=(-1/sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x + sqrt(x^2-4*x))/2)):
a := (2 - x + sqrt(x^2 - 4*x))/2:
b := (2 - x - sqrt(x^2 - 4*x))/2:
P := n->A*a^(n-1)+B*b^(n-1):
R := k->(-2+x)^2*(3-5*x+x^2)*P(2*k-2)-(8-14*x+7*x^2-x^3)*P(2*k-3):
L := 'factor(convert(series(R(n),x,2*n+3),polynom))'$n=0..5;
2 2 2 3 2 2
(-2 + x) , (3 - 5 x + x ) (-2 + x) , (x - 1) (x - 8 x + 17 x - 5) (-2 + x) ,
6 5 4 3 2 2
(x - 13 x + 63 x - 140 x + 142 x - 59 x + 7) (-2 + x) ,
8 7 6 5 4 3 2
2
(x - 17 x + 117 x - 418 x + 827 x - 898 x + 502 x - 124 x +
9) (-2 + x) ,
9 8 7 6 5 4
3 2 2
(x - 1) (x - 20 x + 167 x - 753 x + 1979 x - 3050 x + 2635 x
- 1153 x + 214 x - 11) (-2 + x)
so i assume you're interested in the "big" factor:
galois(L[2]/(x-2)^2);
"2T1", {"S(2)"}, "-", 2, {"(1 2)"}
galois(L[3]/(x-2)^2/(x-1));
"3T1", {"A(3)"}, "+", 3, {"(1 2 3)"}
galois(L[4]/(x-2)^2);
"6T16", {"S(6)"}, "-", 720, {"(3 6)", "(1 6)",
"(2 6)", "(4 6)", "(5 6)"}
galois(L[5]/(x-2)^2);
"8T50", {"S(8)"}, "-", 40320, {"(4 8)", "(1 8)", "(7 8)",
"(2 8)", "(5 8)", "(6 8)", "(3 8)"}
galois(L[6]/(x-2)^2/(x-1));
"9T34", {"S(9)"}, "-", 362880, {"(8 9)", "(5 9)", "(6 9)", "(7
9)", "(3 9)", "(4 9)", "(1 9)", "(2 9)"}
this is the limit of Maple's implementation.
Best, Laurent
On 3/18/06, marta asaeda <[EMAIL PROTECTED]> wrote:
> Hello,
>
> I am having a problem: I would like to find galois group for each
> polynomial in a sequence of polynomials parametrized by n. It is given
> by a recursive formula, so it does involve non-polynomials for
> expression in terms of n. I have this ( I'm mixing mathematica and
> gap notations just for this message):
>
> A[x]:=(1/Sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x - Sqrt(x^2
> - 4*x))/2))
> B[x]:=(-1/Sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x + Sqrt(x^2
> - 4*x))/2))
>
> a := (2 - x + Sqrt[x^2 - 4x])/2
> b := (2 - x - Sqrt[x^2 - 4x])/2
>
> P[n_, x_] := A[x]a^(n - 1) + B[x]b^(n - 1)
>
> R[k_, x_] :=
> (-2+x)^2(3-5x+x^2) P[2(k - 1), x] - (8-14x+7x^2-x^3) P[2(k - 1) - 1, x]
>
> R[k,x] is a polynomial for any positive integer k. I would like to
> give the list of galois groups for each k, say, 5<k<100, or 1000, just
> as much as it is doable by gap in a few days. If I just set it at
> night, go to bed, and see 1000 of galois groups spitted out, that will
> be wonderful. However, it seems gap is having problem dealing with
> 1/Sqrt(x^2 - 4*x). I just tried to teach A, B, a, b one by one, so I
> did like
>
> gap> x:=Indeterminate(Rationals);
> gap> Ax:=(1/Sqrt(x^2 - 4*x))*(x^2 - 4*x + 3 - (2 - x)*((2 - x -
> Sqrt(x^2 - 4*x))/2));
>
> then I get error message like
> Error, no method found! For debugging hints type ?Recovery from NoMethodFound
> Error, no 1st choice method found for `Sqrt' on 1 arguments called from
> Error( no_method_found ); called from
> <function>( <arguments> ) called from read-eval-loop
> Entering break read-eval-print loop ...
> you can 'quit;' to quit to outer loop, or
> you can 'return;' to continue
>
>
> Could anyone please tell me what I should do ?
>
> Thank you ~
>
> marta
>
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--
Laurent Bartholdi \ laurent.bartholdi<at>gmail<dot>com
EPFL SB SMA IMB MAD \ Téléphone: +41 21-6935458
Station 8 \ Secrétaire: +41 21-6935501
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