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Mathematica offers the package FiniteFields, which supports generation of an irreducible polynomial in a finite field:

IrreduciblePolynomial[s,p,d]: gives an irreducible polynomial in the symbol s of degree d over the integers modulo the prime p.

I have two issues with this function:

  1. The polynomial generated in this way is always fixed. How can I produce various irreducible polynomials?
  2. The finite field with respect to which the irreducible polynomial is generated is GF(p), which is of prime order. How can I generate irreducible polynomial with respect to GF($p^n$), for $n>1$?

Edit: An example of what I'm seeking is available here: http://theory.cs.uvic.ca/gen/poly.html

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Rabin's test of irreducibility may help. See at the bottom of wikipedia's page en.wikipedia.org/wiki/… –  tchronis Jul 14 '13 at 10:44
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