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

Let F be a field. Prove that there are infinitely many irreducible monic polynomials
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SchepperJ
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 $a\in F$, x-a is monic and irreducible, as its degree is 1. The set $\left\{x-a,a\in F\right\}$ 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 $F={F}_{q}$, the finite field with q elements.
Let n be a positive integer.
Then $\mathrm{\exists }$ a finite field extension ${F}_{{q}^{n}}$ of degree n.
So, ${F}_{{q}^{n}}={F}_{q}\left(\alpha \right)$, where $\alpha$ satisfies a monic irreducible polynomial ${f}_{n}\left(X\right)\in F\left[x\right].$
Then the set $\left\{{f}_{n}\left(x\right):n\ge 1\right\}$ is an infinite set of monic irreducile polynomials over F.