“Cyclotomic Polynomial”的概念、定义、习题解题方法及翻译-数学辞典

单词

Cyclotomic Polynomial

释义

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 anirreducible 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 firstcyclotomic polynomial to have a coefficient other than

and 0. This is true because 105 is the first number to havethree 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

, and the number of terms having

. Furthermore, assume

, then the middle Coefficient of

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.

See also Aurifeuillean Factorization, Möbius Inversion Formula

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 A013594in ``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.

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