Hi,
I am working on a package for Galois Fields. I nearly finished coding the
functions but unfortunately at the moment most of them are not documented.
Having Maxima 5.28 available you can do the following
(%i1) display2d : false$
(%i2) gf_irr_p(x^4+x+1, 2);
(%o2) true
which means that x^4+x+1 is irredicible over F2,
while it is not over F3:
(%i3) gf_irr_p(x^4+x+1, 3);
(%o3) false
(%i4) gf_factor(x^4+x+1, 3);
(%o4) (x+2)*(x^3+x^2+x+2)
How to compute all irredicible polynomials over e.g. F2 up to a certain
degree:
First set an arbitrary field over F2:
(%i5) gf_set(2,4);
(%o5) [x,x^4+x+1]
This sets F2^4 (primitive element:x, poly for reduction: x^4+x+1) and tells
gf_n2p which converts a number to the corresponding polynomial to use the
modulus 2. (It might be a good idea to allow the modulus to be an optional
parameter to gf_n2p in the future. Then setting a field wouldn't be
necessary.)
(%i6) for n:1 thru 2^6-1 do (p : gf_n2p(n), if gf_irr_p(p) then print(p));
x
x+1
x^2+x+1
x^3+x+1
x^3+x^2+1
x^4+x+1
x^4+x^3+1
x^4+x^3+x^2+x+1
x^5+x^2+1
x^5+x^3+1
x^5+x^3+x^2+x+1
x^5+x^4+x^2+x+1
x^5+x^4+x^3+x+1
x^5+x^4+x^3+x^2+1
(%o6) done
These are all irredicible polynomials over F2 of degree 1 through 5.
Hope that helps
Volker van Nek
2012/10/25 Sara Mussie <saramussie at gmx.net>
> Dear members,
>
> hope all is well with you.
>
> Can I get all irreducible polynomials of any degree with Maxima? If yes,
> can you please tell me how?
>
> Thank you.
>
> Regards,
> Sara
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