Irreducible polynomials of degree greater than 4 over finite fields I want to build a field with $p^{n}$ elements. I know that this can be done by finding a irreducible (on $Z_{p}$) polynomial f of degree n and the result would be the $Z_{p}$/f.
 My question is finding this irreducible polynomial. I know that if it has degree $\leq$ 3, then it's irreducible iff it has no roots. But what if I want to construct a field with 81 = $3^{4}$ elements? How can I find an irreducible polynomial of degree 4?

Find the irreducible quadratics. Multiply them together. Those fourth degree polynomials won't do. Now try some others at random (or systematically, following a list in some natural order). When you find one with no roots you're done.
This is mildly tedious, but you'll get good at the arithmetic, which may come in handy in other computations in the future.
You can also ask Wolfram alpha to factor polynomials modulo $3$.
