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

Isa Trevino

Isa Trevino

Answered question

2020-10-23

F should be a field. Show that there are infinitely many monic polynomials that are irreducible.

Answer & Explanation

SchepperJ

SchepperJ

Skilled2020-10-24Added 96 answers

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 aF, x-a is monic and irreducible, as its degree is 1. The set {xa,aF} 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=Fq, the finite field with q elements.
Let n be a positive integer.
Then a finite field extension Fqn of degree n.
So, Fqn=Fq(α), where α satisfies a monic irreducible polynomial fn(X)F[x].
Then the set {fn(x):n1} is an infinite set of monic irreducile polynomials over F.

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