Building/Working in Finite Field Extensions

["Mills, D. DR MATH" <ad3943@exmail.usma.army.mil>]

Here is what I am trying to do:

(1) Build the finite field GF(p^n) from the prime field GF(p) using a
primitive polynomial of degree n over GF(p) (easy enough);
(2) Build GF(p^(nm)) from GF(p^n) using a primitive polynomial of degree m
over GF(p^n) (not so easy!). 

I am performing the second task (rather than just build GF(p^(nm)) from
GF(p) with a primitive polynomial in GF(p)[x] of degree nm) because I need
to manipulate polynomials of degree m over GF(p^n), specifically factor them
over GF(p^n) as well as use roots of irreducible polynomials of degree m
over GF(p^n). It appears I can't define a root u of such a polynomial as

u=Mod(x,<polynomial of degree m over GF(p^n)>)

unless said polynomial's coefficients come from GF(p), which will usually
not be the case. I'm sure I am missing something obvious here, and would
appreciate it if someone could help me see the forest for the trees. :-)

Thanks.

-Don Mills