Cyclotomic polynomials without using the zeros of $Y^n-1$

TL;DR

The paper addresses the traditional reliance on a splitting field for $Y^n-1$ to establish the irreducibility of the $n$-th cyclotomic polynomial $\Phi_n(Y)$. It introduces an intrinsic construction with $\Psi_n$ as $\mathrm{lcm}_{0<d<n, d\mid n}(Y^d-1)$ and $\Phi_n(Y)=(Y^n-1)/\Psi_n(Y)$, yielding the field $\mathbb{Q}_n=\mathbb{Q}[Y]/\langle \Phi_n(Y)\rangle$ as an implicit splitting field. The main contributions are a complete, constructive proof of irreducibility, a clear factorization $Y^n-1=\prod_{d\mid n}\Phi_d(Y)$ that does not invoke roots of unity, and a connection $\mathbb{Q}[\xi]\cong\mathbb{Q}_n$ when $\xi$ is a primitive $n$-th root of unity. The approach aligns with Gauss, Kummer, Kronecker, and Bishop's constructive tradition and provides an avenue for formalization in constructive systems, offering an alternative foundational perspective on cyclotomic polynomials.

Abstract

This note aims to construct an ``intrinsic'' splitting field for the polynomial $Y^n-1$ over the rational field $\bf Q$, in a way that Gauss, Kummer, Kronecker and Bishop would have liked. Contrary to the usual presentations, our construction does not use any splitting field of $Y^n-1$ which would be given before demonstrating the irreducibility of the cyclotomic polynomial.

Theorems & Definitions (30)

• Proposition 1.1: $\mathbb {Z}[Y]$ is a gcd domain
• Lemma 1.2
• Lemma 1.3: Substitution Lemma
• proof
• Lemma 1.4: Roots and Decomposition
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof