Cyclotomic Polynomial

A polynomial given by

 (1)

where are the primitive th Roots of Unity in given by . The numbers are sometimes called de Moivre Numbers. is an irreducible Polynomial in with degree , where is the Totient Function. For Prime,
 (2)

i.e., the coefficients are all 1. has coefficients of for and , making it the first cyclotomic polynomial to have a coefficient other than and 0. This is true because 105 is the first number to have three distinct Odd Prime factors, i.e., (McClellan and Rader 1979, Schroeder 1997). Migotti (1883) showed that Coefficients of for and distinct Primes can be only 0, . Lam and Leung (1996) considered
 (3)

for Prime. Write the Totient Function as
 (4)

and let
 (5)

then
1. Iff for some and ,

2. Iff for and ,

3. otherwise .
The number of terms having is , and the number of terms having is . Furthermore, assume , then the middle Coefficient of is .

The Logarithm of the cyclotomic polynomial

 (6)

is the Möbius Inversion Formula (Vardi 1991, p. 225).

The first few cyclotomic Polynomials are

The smallest values of for which has one or more coefficients , , , ... are 0, 105, 385, 1365, 1785, 2805, 3135, 6545, 6545, 10465, 10465, 10465, 10465, 10465, 11305, ... (Sloane's A013594).

The Polynomial can be factored as

 (7)

where is a Cyclotomic Polynomial. Furthermore,
 (8)

The Coefficients of the inverse of the cyclotomic Polynomial
 (9)

can also be computed from
 (10)

where is the Floor Function.

References

Beiter, M. The Midterm Coefficient of the Cyclotomic Polynomial .'' Amer. Math. Monthly 71, 769-770, 1964.

Beiter, M. Magnitude of the Coefficients of the Cyclotomic Polynomial .'' Amer. Math. Monthly 75, 370-372, 1968.

Bloom, D. M. On the Coefficients of the Cyclotomic Polynomials.'' Amer. Math. Monthly 75, 372-377, 1968.

Carlitz, L. The Number of Terms in the Cyclotomic Polynomial .'' Amer. Math. Monthly 73, 979-981, 1966.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, 1996.

de Bruijn, N. G. On the Factorization of Cyclic Groups.'' Indag. Math. 15, 370-377, 1953.

Lam, T. Y. and Leung, K. H. On the Cyclotomic Polynomial .'' Amer. Math. Monthly 103, 562-564, 1996.

Lehmer, E. On the Magnitude of Coefficients of the Cyclotomic Polynomials.'' Bull. Amer. Math. Soc. 42, 389-392, 1936.

McClellan, J. H. and Rader, C. Number Theory in Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1979.

Migotti, A. Zur Theorie der Kreisteilungsgleichung.'' Sitzber. Math.-Naturwiss. Classe der Kaiser. Akad. der Wiss., Wien 87, 7-14, 1883.

Schroeder, M. R. Number Theory in Science and Communication, with Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, 3rd ed. New York: Springer-Verlag, p. 245, 1997.

Sloane, N. J. A. Sequence A013594 in The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Vardi, I. Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, pp. 8 and 224-225, 1991.