Galois field

[Mikael Olofsson]

Roie Kerstein wrote:
> 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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