Gauss's Cyclotomic Formula
Let 
be a prime number, then
where 
and 
are homogeneous polynomials in 
and 
with integer coefficients. Gauss (1965, p. 467) gives the coefficients of 
and 
up to 
.
Kraitchik (1924) generalized Gauss's formula to odd squarefree integers 
. Then Gauss's formula can be written in the slightly simpler form
where 
and 
have integer coefficients and are of degree 
and 
, respectively, with 
the totient function and 
a cyclotomic polynomial. In addition, 
is symmetric if 
is even; otherwise it is antisymmetric. 
is symmetric in most cases, but it antisymmetric if 
is of the form 
(Riesel 1994, p. 436). The following table gives the first few 
and 
s (Riesel 1994, pp. 436-442).
REFERENCES:
Gauss, C. F. §356-357 in Untersuchungen über höhere Arithmetik. New York: Chelsea, pp. 425-428 and 467, 1965.
Kraitchik, M. Recherches sue la théorie des nombres, tome I. Paris: Gauthier-Villars, pp. 93-129, 1924.
Kraitchik, M. Recherches sue la théorie des nombres, tome II. Paris: Gauthier-Villars, pp. 1-5, 1929.
Riesel, H. "Gauss's Formula for Cyclotomic Polynomials." In tables at end of Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 436-442, 1994.
