Thanks to Malcolm Greig, here is a table of primitive polynomials for building GF(p^{n}) for all p^{n} up to 2^{30}, with n > 1.
I have translated this
table
in Maple format through a characteristic by characteristic index.
The uncompressived file is about 133Kb in size.
Only one primitive polynomial is given for each order.
This table was used in generating combinatorial designs,
but doubtless has other uses.
The theory used in compiling the table was taken from:
J. D. Alanen and D. E. Knuth, Tables of finite fields, Sankhya A (1964) 26, 305328.
The file is in fixed format (actually (I10,I7,I3,30I6) for those who remember FORTRAN).
The fields are v,p,n,b_{n1},b_{n2},...,b_{0} and are ordered within the file
by p, then n within p, where v=p^{n}. Each p^{n} is on a separate line.
The primitive polynomial is
f(x)=x^{n}+b_{n1}x^{n1}+b_{n2}x^{n2}+...+b_{0}
The leading term x^{n} is implicit. This polynomial has the property that the successive powers
of any one of the roots of f(x)=0 generate GF(p^{n}).
Also, if n is even, then successive powers of b_{0} generate GF(p),
whilst if n is odd, then successive powers of pb_{0} generate GF(p).
I no longer remember precisely what criteria I used in selecting the polynomial;
I do remember it was similar to Alanen and Knuth's criteria, but was not identical.
It does have the property that most coefficients are zero, and those that
are nonzero tend to be ones (except b_{0}, of course).

Malcolm Greig

