Exploit cyclicity of roots-of-unity groups to simplify the irreducibility proof of cyclotomic polynomials

Develop a proof of the irreducibility of the cyclotomic polynomial Φ_n(Y) over the rational field Q that relies directly on the fact that the group of roots of unity in any field is cyclic, thereby yielding a significantly simpler argument than the one presented in this work, which avoids using splitting fields of Y^n−1.

Background

The paper gives an intrinsic, constructive development of cyclotomic polynomials Φ_n(Y) without invoking the prior existence of a splitting field for Y^n−1. It proves standard properties, including irreducibility over Q, via gcd/lcm arguments and substitution lemmas, avoiding explicit use of primitive roots of unity until the end.

In the conclusion, the authors note that while it is easy to prove that the group of roots of unity in a field is cyclic, they were not able to leverage this fact to obtain a much simpler proof of irreducibility. This leaves open whether a streamlined proof can be built using only this cyclicity property within their intrinsic framework.