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 \(\displaystyle{\left\lbrace{x}-{a},{a}\in{F}\right\rbrace}\) 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 \(\displaystyle\exists\) 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 \(\displaystyle{\left\lbrace{{f}_{{n}}{\left({x}\right)}}:{n}\ge{1}\right\rbrace}\) is an infinite set of monic irreducile polynomials over 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 \(\displaystyle{\left\lbrace{x}-{a},{a}\in{F}\right\rbrace}\) 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 \(\displaystyle\exists\) 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 \(\displaystyle{\left\lbrace{{f}_{{n}}{\left({x}\right)}}:{n}\ge{1}\right\rbrace}\) is an infinite set of monic irreducile polynomials over F.