Another problem about irreducible polynomials over a (finite) field

Question

I want to know whether it is true that over a finite field $K$ (with characteristic $p$, say), and for any positive integer $m$, does there always exist a prime (or equivalently, irreducible, since the polynomial ring over a field is UFD) polynomial in $K[x]$ with degree $m$. I prefer some elementary proof.

Answer 1

Yes, you can. There is a formula for the number of monic primes of degree $m$ over a finite field of order $q$:

$$\nu_q(m)=\frac{1}{m}\sum_{d\mid m} \mu\left(\frac md\right)q^d$$

Where $\nu_q(m)$ counts the number of prime monic polymomials of degree $m$ over a field of size $q$.

You can easily show this gives $\nu_q(m)\geq \frac{1}{m}\left(q^m - \sum_{i=0}^{m-1} q^i\right)> 0$.