Gauss's cyclotomic formula expresses a quotient of two 
th
powers in terms of homogeneous
polynomials. 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, the
coefficients of 
are symmetric if its degree 
is even and antisymmetric otherwise. The coefficients
of 
are antisymmetric when 
is composite and of
the form 
, and symmetric otherwise (Brent 1993, p. 139).
The following table gives the first few 
and 
s (Riesel 1994, pp. 436-442).
 |  |  |
| 5 |  | 1 |
| 7 |  |  |
| 11 |  |  |
