Irreducible Polynomials p 5 degree 4

[grandnexus]

I am attempting to construct a field containing 625 elements and should be in the form Zn[x] mod f(x).

Factoring 625 leads to 5^4. So I'm guessing my field will be GF(5^4). So in order for me to construct a field with all elements in it, I need f(x) to be some irreducible polynomial mod 5 of degree 4.

How can I go about finding irreducible polynomials? I know I can choose all the polynomials below degree 4 with coefficients mod 5 and attempt to find one without factors, but that would take forever. Is there a quick way to do this or a list of precomputed irreducible polynomials given GF(p^n) where p is prime and n is greater than 1??

Thanks.

 

[matt grime]

I can't imagine anyone's bothered to precompute anything. The solution is to think about F_5, the field with 5 elements, and decide if you know any polynomial that every element of F_5 satisfies.