Several Proofs of the Irreducibility of the Cyclotomic Polynomials

We present a number of classical proofs of the irreducibility of the n-th cyclotomic polynomial Φn(x). For n prime we present proofs due to Gauss (1801), in both the original and a simplified form, Kronecker (1845), and Schönemann/Eisenstein (1846/1850), and for general n proofs due to Dedekind (1857), Landau (1929), and Schur (1929). Let Φn(x) denote the n-th cyclotomic polynomial, defined by Φn(x) = ∏(x−ζ ) where ζ ranges over the primitive n-th roots of unity. Φn(x) is also given inductively by Φn(x) = xn −1 ∏Φd(x) where d ranges over the proper divisors of n. In case n = p is prime, Φp(x) = (xp−1)/(x− 1) = xp−1 + xp−2 + · · ·+ x+1. It is a basic result in number theory that Φn(x) is irreducible for every positive integer n. It is our objective here to present a number of classical proofs of this theorem (certainly not all of them). The irreducibility of Φp(x) for p prime was first proved by Gauss [4, article 341], with a simpler proof being given by Kronecker [5] and even simpler and more general proofs being given by Schönemann [8] and Eisenstein [3]. We give these proofs here. (The last of these has become the standard proof.) Gauss’s proof is rather complicated, so we also give a simpler proof along the same lines. The irreducibility of Φn(x) in general was first proved by Kronecker [6], with simpler proofs being given by Dedekind [2], Landau [7], and Schur [9]. We give the last three of these proofs here. (A variant of Dedekind’s proof has become the standard proof.) With the exception of Schur’s proof, which uses some results about algebraic integers, these proofs all just use basic results about polynomials. We have organized this paper to collect background material about polynomials in a preliminary section, to have it available when we present the main results.