computer algebra system for polynomial algebras over finite fields

You can do these things in Sage quite easily. Here is an example (using Sage 5.2):

sage: Fp.<mu>=GF(5)[]
sage: Fp
Univariate Polynomial Ring in mu over Finite Field of size 5
sage: m=mu^5-mu+1
sage: K.<y>=GF(5^5, name='y', modulus=m) # your mu becomes y
sage: A.<x>=K[]
sage: A
Univariate Polynomial Ring in x over Finite Field in y of size 5^5
sage: f=x^7-1
sage: f(x+y)
x^7 + 2*y*x^6 + y^2*x^5 + (y + 4)*x^2 + (2*y^2 + 3*y)*x + y^3 + 4*y^2 + 4
sage: g=x^2+x+1
sage: f(x+y)+g(x)
x^7 + 2*y*x^6 + y^2*x^5 + y*x^2 + (2*y^2 + 3*y + 1)*x + y^3 + 4*y^2

Another option is to use GAP (kindly provided by A.Konovalov)

gap> R:=PolynomialRing(GF(5),"mu"); mu:=Indeterminate(GF(5));;
GF(5)[mu]
gap> m:=mu^5-mu+1;
mu^5-mu+Z(5)^0
gap> T:=AlgebraicExtension(GF(5),m); a:=RootOfDefiningPolynomial(T);;
<field of size 3125>
gap> A:=PolynomialRing(T,"x"); x:=Indeterminate(T);;
<object>[x]
gap> f:=x^7-1;
x^7-!Z(5)^0
gap> Value(f,x+a);
x^7+Z(5)*a*x^6+a^2*x^5+(a-Z(5)^0)*x^2+(Z(5)*a^2+Z(5)^3*a)*x+(a^3-a^2-Z(5)^0)
gap> g:=x^2+x+One(T); 
x^2+x+!Z(5)^0 
gap> Value(f,x+a)+g; 
x^7+Z(5)*a*x^6+a^2*x^5+a*x^2+(Z(5)*a^2+Z(5)^3*a+Z(5)^0)*x+(a^3-a^2)

in maxima:

 f(x):= x^3+x+5;
 modulus:5;
 algebraic:true;
 rat(f(u)); 

returns 2*u+1

In addition, Magma http://magma.maths.usyd.edu.au/magma/ and GAP http://www.gap-system.org/ will perform these computations. The former is commercial and the latter is free. If you want to compute over large finite fields, then you may want to try Magma.

Stephen Glasby