On the Middle Coefficient of a Cyclotomic Polynomial

Abstract

Clearly (, has degree 0 (n), where 0 signifies Euler's totient function. These monic polynomials can be defined recursively as (I1 (x) = x 1 and Hiln D (x) = xn 1 for n > 1. The first few are easily calculated to be x 1, x + 1, x2 + x + , x2 + 1.... For these and other basic facts, see an algebra text such as [5]. While it might appear that the coefficients of the cyclotomic polynomials are always ?1, the presence of 2x7 in o105(x) shows that this is not invariably the case (and indeed is a good counterexample for those students who insist that the law of small numbers is universally valid; see [4] for further discussion). Naturally, much work has been done on the values of the coefficients of On (x). One amazing fact worthy of mention is that every integer appears as a coefficient in some cyclotomic polynomial (see [1], [8]). In this article, we provide a short and elementary proof of the following result: