Let F be a field. Prove that there are infinitely many irreducible monic polynomials

To prove the existence of infinitely many monic irreducible polynomials over any given field F.
A polynomial f(x) in F[x] is irreducible if f(x) cannot be factorized as f(x)=g(x)h(x), with g,h in F[x], ie coefficients in F. and deg g and h both greater than 1. In other words, f(x) is irreducbile if it does not be written as a non-trivial product.
If F is an infinite field, there is really nothing to prove: given any ${\displaystyle a\in F}$, x-a is monic and irreducible, as its degree is 1. The set ${\displaystyle \{x-a,a\in F\}}$ is therefore an infinite set of monic irreducible polynomials
So the problem is mainly for the case when F is a finite field. In this case, the argument is as shown.
Let F be a finite field, say ${\displaystyle F=F_q}$, the finite field with q elements.
Let n be a positive integer.
Then ${\displaystyle \mathrm{\exists }}$ a finite field extension ${\displaystyle F_{q^n}}$ of degree n.
So, ${\displaystyle F_{q^n}=F_q\left(\alpha \right)}$, where ${\displaystyle \alpha }$ satisfies a monic irreducible polynomial ${\displaystyle f_n\left(X\right)\in F\left[x\right].}$
Then the set ${\displaystyle \{f_n\left(x\right):n\ge 1\}}$ is an infinite set of monic irreducile polynomials over F.