Cyclotomic and double angle polynomials

Abstract

One admires and applauds the enterprise of anyone who uses Gauss’s 1801 Disquisitiones arithmeticae as the starting point for mathematical exploration. I enjoyed McKeon and Sherry’s description of their journey [1] and the challenge of their conjectures. They drew attention to a class of polynomials that satisfy what they called the double angle condition ((1) below). Unfortunately, their failure to work with an appropriate definition of cyclotomic polynomials seriously handicapped their computer-aided attempt to classify double angle polynomials. Once this is remedied, a pleasant classification emerges, at least for polynomials with rational coefficients, without recourse to a computer. The main aim of this article is to present this classification. A brief final section considers McKeon and Sherry’s conjectures about irreducible double angle polynomials.