Take the 2-minute tour ×  Suppose $F$ is a field s.t
$\left|F\right|=q$. Take $p$ to be some prime. How many monic
irreducible polynomials of degree $p$ can exist over $F$?
share|improve this question What have you tried? (Think about
subextensions of $\mathbb{F}_{q^p}$ over $\mathbb{F}_q$.) – 
Qiaochu Yuan May 23 '11 at 9:31 @Qiaochu: Thanks, got the idea
and found the answer... will write it down here soon! –  IBS May
23 '11 at 13:25  2 Answers 2  up vote 22 down vote accepted  The
number of such polynomials is exactly $\displaystyle
\frac{q^{p}-q}{p}$ and this is the proof:  The two main facts
which we use (and which I will not prove here) are that
$\mathbb{F}_{q^{p}}$ is the splitting field of the polynomial
$g\left(x\right)=x^{q^{p}}-x$, and that every monic irreducible
polynomial of degree $p$ divides $g$.  Now:
$\left|\mathbb{F}_{q^{p}}:\mathbb{F}_{q}\right|=p$ and therefore
there could be no sub-extensions. Therefore, every irreducible
polynomial that divides $g$ must be of degree $p$ or 1. Since ea