Irreducible cubic polynomial in char 2
- From: ema <emanuele.cesena@xxxxxxxxx>
- Date: Wed, 2 Apr 2008 07:20:02 -0700 (PDT)
Hi,
I have a polynomial F = x^3 + x + k, with k in \F_q, q=2^m (m prime).
I know that F is irreducible, and I would like to find a root in a
degree 3 extension of \F_q.
Well, if I take L = \F_q[x]/(F), then x is a solution.
However, I prefer to consider L=\F_q[a]/(a^3+a+1).
So, I'd like to find x_i, s.t. x = x_2 a^2 + x_1 a + x_0 and F(x)=0.
It is easy to prove that x_0 = 0.
Any idea on how to explicitly find x_1/x_2?
It would be nice to find them as function of k... please note that I
need only a single solution.
I'm looking at Bill Allombert "Explicit Computation of Isomorphisms
Between Finite Fields", Finite Fields, 8, 2002, 332-342.
http://www.math.u-bordeaux.fr/~allomber/fpisom.ps
but at the moment I have no useful result.
Thank you in advance,
--
ema
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