Thank you very much!
However, I lack two important things in the packages introduced there:
1. There is only element arithmetic - no polynomials, in which I have need.
2. there is support only for fields of order 2^p, while I need fields of
prime order.
You still helped me, as before I searched for galois fields and found
nothing, and search for finite fields yields more results.
Off topic for the list, but possibly on topic for this thread:
At our department we use Magma (http://magma.maths.usyd.edu.au/) for finite field arithmetic and error control codes. Magma has nothing to do with Python, instead it is a very mature tool of its own, mainly for descrete math. It knows what a permutation group is, it knows what GF(81) is, and much more. Check out the handbook (Magma on-line help) on the web site and be impressed. You will find finite fields under Basic Rings. Magma is not free, though, and licence fees vary depending on who is buying the licence, see the web site for details. Note that under certain circumstances you may be eligible for a free version. If you have access to Magma where you are, do consider using it.
An example Magma session:
banin:~> magma Magma V2.8-9 Fri Dec 3 2004 13:17:54 on banin [Seed = 735720219] Type ? for help. Type <Ctrl>-D to quit.
R<x>:=PolynomialAlgebra(GF(3)); F<a>:=GF(27); for el in F do
for> print el, " ", MinimalPolynomial(el); for> end for; 1 x + 2 a x^3 + 2*x + 1 a^2 x^3 + x^2 + x + 2 a^3 x^3 + 2*x + 1 a^4 x^3 + x^2 + 2 a^5 x^3 + 2*x^2 + x + 1 a^6 x^3 + x^2 + x + 2 a^7 x^3 + x^2 + 2*x + 1 a^8 x^3 + 2*x^2 + 2*x + 2 a^9 x^3 + 2*x + 1 a^10 x^3 + x^2 + 2 a^11 x^3 + x^2 + 2*x + 1 a^12 x^3 + x^2 + 2 2 x + 1 a^14 x^3 + 2*x + 2 a^15 x^3 + 2*x^2 + x + 1 a^16 x^3 + 2*x + 2 a^17 x^3 + 2*x^2 + 1 a^18 x^3 + x^2 + x + 2 a^19 x^3 + 2*x^2 + x + 1 a^20 x^3 + 2*x^2 + 2*x + 2 a^21 x^3 + x^2 + 2*x + 1 a^22 x^3 + 2*x + 2 a^23 x^3 + 2*x^2 + 1 a^24 x^3 + 2*x^2 + 2*x + 2 a^25 x^3 + 2*x^2 + 1 0 x
Regards /Mikael Olofsson Universitetslektor (Senior Lecturer [BrE], Associate Professor [AmE]) Linköpings universitet
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