In a finite field there exists an irreducible polynomial of degree at least $n$ for $n \in \mathbb{N}$

This question already has an answer here, but I'm looking for a solution that doesn't use field extensions. It's relatively easy to find a polynomial without zeros for every $$n$$ but as far as I know this doesn't imply that the polynomial is irreducible (maybe in a finite field it does?). Thank you for your answers.

Let $$F$$ be a finite field of characteristic $$p$$ and let $$n\in\Bbb{N}$$. Let $$k$$ be an integer coprime to $$p$$. Then the polynomial $$x^k+1$$ is separable over $$F$$, so it is a product of distinct irreducible factors. For any degree $$d$$ there are only finitely many (irreducible) polynomials of degree $$d$$ with coefficients in $$F$$, so for sufficiently large values of $$k$$ it must have an irreducible factor of degree $$n$$.
Hint use Euclid's idea: consider $$1+p_1\cdots p_k$$ where the $$p_i$$ are all irreducibles of degree $$< n$$