Re: [Axiom-math] Working with finite fields

[Martin Rubey]

Dear Humberto,

I'm not quite sure whether this is what you were looking for or not, but maybe
it helps:

reduce(+,[coordinates(a^5128).i *q^(i-1) for i in 1..#coordinates (a^5128)])

I hope HyperDoc works for you!

Martin

Humberto Ortiz-Zuazaga writes:
 > I've got some code I'm converting from Macaulay2 to Axiom that deals
 > with finite fields. I have a question regarding converting fields to
 > other representations
 > 
 > In maculay, I can set up a field with a particular quotient ring:
 > 
 > -- Set up a 13 bit polynomial ring over Z_2
 > 
 > i1 : A = ZZ/2[X]/(X^13+X^8+X^5+X^3+X^2+X+1)
 > 
 > o1 = A
 > 
 > o1 : QuotientRing
 > 
 > -- Make the Galois Field, a is the root
 > 
 > i2 : k = GF(A,Variable => a)
 > 
 > o2 = k
 > 
 > o2 : GaloisField
 > 
 > -- This uses a representation like Axiom's FiniteFieldCyclicGroup
 > 
 > i5 : a^5128
 > 
 >       5128
 > o5 = a
 > 
 > o5 : k
 > 
 > -- But I can look at the term like a polynomial also:
 > 
 > i6 : lift(a^5128, ambient k)
 > 
 >       11    10    9    5    4    2
 > o6 = X   + X   + X  + X  + X  + X
 > 
 > o6 : A
 > 
 > I can do everything except the last step in Axiom:
 > 
 > 
 >  m := monomial(1,1)$SUP(GF2)
 >  f := m^13+m^8+m^5+m^3+m^2+m+1
 >  GF8192 := FFCGP(GF2,f)
 >  a := primitiveElement()$GF8192
 > 
 > So now:
 > 
 > (6) -> a**5128
 > 
 >           5128
 >    (6)  %A
 > Type: FiniteFieldCyclicGroupExtensionByPolynomial(PrimeField 
 > 2,?**13+?**8+?**5+?**3+?*?+?+1)
 > 
 > but I can't understand the output of )display op lift to figure out how
 > to get the result I want. I'm building the February (patch-30) release
 > on Ubuntu Warty as we speak, in the hope that Browse/Hypertex will make
 > it easier to understand the available documentation.
 > 
 > I do know I could use the FFPX constructor to get the large field in
 > polynomial form, and then call discreteLogarithm to get the exponents. I
 > guess my question is can I do the converse easily?
 > 
 > -- 
 > Humberto Ortiz Zuazaga
 > Programmer Archaeologist
 > University of Puerto Rico
 > High Performance Computing facility
 > http://www.hpcf.upr.edu/~humberto/
 > 
 > 
 > _______________________________________________
 > Axiom-math mailing list
 > address@hidden
 > http://lists.nongnu.org/mailman/listinfo/axiom-math