I want to prove that given K a finite field and n>0, exists an irreducible polynomial f ∈ K[x] of degree n.
I am taking into account this explanation given in Field Theory by Roman: 
But I don't understand the implications given.
My attempt is to use the fact that a finite field extension (finite degree) of a finite field is simple and then work from there but I don't get it. Also, the notation of extension using < confuses me.
Thank you in advance!
It is well-known (i.e. in every introduction text) that $\operatorname{GF}(p^r)$ consists of all the roots of the polynomial $X^{p^r} - X \in \Bbb F_p[X]$ in some algebraic closure of $\Bbb F_p$.
Therefore, $\operatorname{GF}(p^r) \setminus \{0\}$ consists of all the $(p^r-1)$st roots of unity, and a primitive $(p^r-1)$st root of unity $\zeta$ exists by a simple combinatorial argument, so $\operatorname{GF}(p^r)$ is generated by $\zeta$ as a field.
Therefore, any (finite) extension of finite fields is simple (just take $\zeta$).