The sextic period polynomial

Abstract

The coefficients of the polynomial whose roots are the six periods of the pth roots of unity are given for every prime p — 6/ 4- 1 in terms of L and M in the quadratic partition 4/? = L2 + 27M2. An explicit formula for the discriminant of this polynomial is also given. A complete analysis of the prime factors of the integers represented by the period polynomial and its corresponding form is given. 1. Introduction. In 1893 Carey [1] developed a method for obtaining the coefficients of the general period polynomial and gave a table of the sextic polynomial for every prime p < 500. His method expresses the coefficients in terms of a sequence {ak}, where ak is the au-element in the kih power of the matrix (/, j) of cyclotomic numbers. It has recently been shown [6] that these α's form a linear recurrence whose scale of relation is the period polynomial and whose initial values are multiple sums of cyclotomic numbers. That Carey's approach to the period polynomial is inefficient is amply demonstrated by the rather long list of errata in Carey's table given in the Appendix to this paper. It is surprising to note that, until now, no one has given explicit formulas for the coefficients of the sextic period polynomial although there are formulas due to Dickson [3] and Whiteman [10] for the corresponding cyclotomic numbers. In this paper we give the coefficients and the discriminant of the sextic period polynomial in terms of the fundamental quadratic partitions