Bill Daly on Fri, 23 Jan 2009 16:43:18 +0100 |
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Re: Polmod factorization |
Denis Simon wrote:
Thanks. This is exactly what I was looking for.Dear Bill, You are probably also interested in the function factornf (see also nffactor). Be careful of the choice of the variables ! For example : gp > f=polcyclo(5) %1 = x^4 + x^3 + x^2 + x + 1 gp > fy=subst(f,x,y); gp > factornf(f,fy) %3 = [x + Mod(-y, y^4 + y^3 + y^2 + y + 1) 1] [x + Mod(-y^2, y^4 + y^3 + y^2 + y + 1) 1] [x + Mod(-y^3, y^4 + y^3 + y^2 + y + 1) 1] [x + Mod(y^3 + y^2 + y + 1, y^4 + y^3 + y^2 + y + 1) 1] Denis SIMON. On Fri, 23 Jan 2009, Karim Belabas wrote:* Bill Daly [2009-01-23 07:10]:If f(x) is an irreducible polynomial in x, then Mod(x,f(x)) is a generic root of f(x), and the algebra mod f(x) is isomorphic (I think) to the algebra of the field generated by appending any root of f(x) to Q. Is there a way of factoring f(x) mod f(x)? What I have in mind is that for some polynomials where Mod(x,f(x)) is a root, then there may be other rational functions of x which are also roots of f(x), e.g. if f(x) is polcyclo(n), then Mod(x^a,f(x)) is a root whenever a is coprime to n. I don't however see any easy way of finding such roots with polmods in PARI. What, if anything, am I overlooking?For general polmods, nothing. On the other hand the formulation suggests that you're actually considering the special case f \in Q[X]. Then you may just use nfgaloisconj(f), which settles the case of Mod(x,f): (09:45) gp > f = polcyclo(5); v = nfgaloisconj(f) %1 = [x, x^2, -x^3 - x^2 - x - 1, x^3]~ \\ the ordering is a bit strange in this case; roots are sorted according to \\ the lexicographic order on Q^deg(f) This settles the case of Mod(x, f); if you're interested in the conjugates of more general Mod(a(x), f), use subst: (09:45) gp > a = x^2 + x; vector(#v, i, lift( subst(a, x, Mod(v[i],f)) )) %2 = [x^2 + x, -x^3 - x - 1, -x^2 - x - 1, x^3 + x] \\ lift() introduced for readability ... Analogous ideas are also implemented over a finite field. Cheers, K.B. -- Karim Belabas, IMB (UMR 5251) Tel: (+33) (0)5 40 00 26 17 Universite Bordeaux 1 Fax: (+33) (0)5 40 00 69 50 351, cours de la Liberation http://www.math.u-bordeaux1.fr/~belabas/ F-33405 Talence (France) http://pari.math.u-bordeaux1.fr/ [PARI/GP] ` Regards, Bill |