An old question that occurred to me again recently: are there any explicit formula known for sequences of irreducible polynomials $g\_{p^n}(X)$ in $Z/pZ[X]$ such that for the finite field with $p^n$ elements


$F\_{p^n} = (Z/pZ)[X] / (g\_{p^n}).$


Such polynomials exist, as anyone who's studied algebra knows, but I've always seen their existence proved nonconsctructively. Ideally, I'm looking for something like


$g\_{p^n}(X) = X^n+c\_{n-1}(p^n)X^{n-1}+...,$


where $c\_{n+1}(p^n)$ is some familiar expression, or something with generating functions, etc.


I'm not an algebraist, so this could have a very obvious answer. One could experiment around with low lying irreducible polynomials to try to come up with a nice pattern, I think, but the fact that I've never seen this presented in an algebra text leads me to suspect that no such formula actually exist and a better answer would come from just asking people in a position to know. In some ways, I realize, it may be like asking for a formula for the nth prime...



Explicit representations of finite fields
There is no known simple explicit formula for an irreducible polynomial
of given degree $n$ over $\mathbb{F}\_p$. However there has been a lot of work
on explicit irreducible polynomials for certain families of $n$, notably
$n$ of the form $rq^s$ where $r$ is fixed and $q$ is a fixed prime. For one
modest contribution to this field see
[this paper](http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WFM-45M8VYC-G&_user=121739&_coverDate=01%2F31%2F1997&_rdoc=1&_fmt=high&_orig=browse&_srch=doc-info%28%23toc%236798%231997%23999969998%23305679%23FLP%23display%23Volume%29&_cdi=6798&_sort=d&_docanchor=&_ct=8&_acct=C000010018&_version=1&_urlVersion=0&_userid=121739&md5=2c4f893bf37032dd8f61bf41c9eda04d) but also see the papers it cites.


