The nth cyclotomic polynomial is defined as follows where .

The polynomial has the following factorization .

In this post, I will give a proof of irreducibility of nth cyclotomic polynomial. I will prove the following theorem.

**Theorem:** is an irreducible polynomial in .

**Proof:** If is not an irreducible polynomial then form Gauss’s lemma it has a non-trivial factorization where .

As has no repeated roots, are relatively prime to each other in . As is a Euclidean domain, there exist functions , and a natural number such that ——–(1).

Without loss of generality, we can assume . Let one root of be where . From Dirichlet’s theorem, there exists a prime number satisfying .

Note that, ——-(2)

Hence, . Therefore, have a common root and therefore they both have a non trivial greatest common divisor in .

Consider these polynomials and as polynomials in $Z_p(x)$. As, in . We have, and have non-trivial common divisor in –which implies and have a non trivial common divisor, , in . This implies in , which is absurd.