TITLE: computer algebra system for polynomial algebras over finite fields
QUESTION [8 upvotes]: Is there a computer algebra system that can do arithmetic over polynomial algebras over finite fields where I can specify the extension?
Exempli gratia, if $f(x), g(x) \in \mathbb{F}_p[\mu]/(m(\mu))[x]$, I'd like the CAS to be able to compute things like $f(x + \mu) + g(x)$ where I specify the polynomial $m(\mu)$.
Thanks

REPLY [2 votes]: Here's one way to do it in Macaulay2
restart
R = GF(5)[mu]/(mu^5-mu+1)[x]
f = x^7-1
g = x^2+x+1
sub(f, x => x+mu) + g

returning
      7        6     2 5       2       2                 3     2
o4 = x  + 2mu*x  + mu x  + mu*x  + (2mu  - 2mu + 1)x + mu  - mu

o4 : R

REPLY [2 votes]: In Singular:
> ring R = (5,mu),(x),dp; minpoly=mu^5-mu+1;
> poly f = x^7 - 1;
> poly g = x^2 + x + 1;
> subst(f,x,x+mu)+g;
x^7+(2*mu)*x^6+(mu^2)*x^5+(mu)*x^2+(2*mu^2-2*mu+1)*x+(mu^3-mu^2)