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Cyclotomic polynomial : ウィキペディア英語版
Cyclotomic polynomial
In mathematics, more specifically in algebra, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients, which is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are the ''n''th primitive roots of unity

e^}
, where ''k'' runs over the integers lower than ''n'' and coprime to ''n''. In other words, the ''n''th cyclotomic polynomial is equal to
:
\Phi_n(x) =
\prod_\stackrel
\left(x-e^}\right)

It may also be defined as the monic polynomial with integer coefficients, which is the minimal polynomial over the field of the rational numbers of any primitive ''n''th-root of unity ( e^ is an example of such a root).
Another important equation, linking the cyclotomic polynomials and primitive roots of unity, is the following one.
:x^n - 1 =\prod_ \left(x- e^ \right)= \prod_ \prod_ \left(x- e^ \right) =\prod_ \Phi_(x) = \prod_ \Phi_d(x)
==Examples==

If ''n'' is a prime number then
:~\Phi_n(x) = 1+x+x^2+\cdots+x^=\sum_^ x^i.
If ''n''=2''p'' where ''p'' is an odd prime number then
:~\Phi_(x) = 1-x+x^2-\cdots+x^=\sum_^ (-x)^i.
For ''n'' up to 30, the cyclotomic polynomials are:
:~\Phi_1(x) = x - 1
:~\Phi_2(x) = x + 1
:~\Phi_3(x) = x^2 + x + 1
:~\Phi_4(x) = x^2 + 1
:~\Phi_5(x) = x^4 + x^3 + x^2 + x +1
:~\Phi_6(x) = x^2 - x + 1
:~\Phi_7(x) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
:~\Phi_8(x) = x^4 + 1
:~\Phi_9(x) = x^6 + x^3 + 1
:~\Phi_(x) = x^4 - x^3 + x^2 - x + 1
:~\Phi_(x) = x^ + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
:~\Phi_(x) = x^4 - x^2 + 1
:~\Phi_(x) = x^ + x^ + x^ + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
:~\Phi_(x) = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
:~\Phi_(x) = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1
:~\Phi_(x) = x^8 + 1
:~\Phi_(x) = x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
:~\Phi_(x) = x^6 - x^3 + 1
:~\Phi_(x) = x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
:~\Phi_(x) = x^8 - x^6 + x^4 - x^2 + 1
:~\Phi_(x) = x^ - x^ + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
:~\Phi_(x) = x^ - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
:~\Phi_(x) = x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
:~\Phi_(x) = x^8 - x^4 + 1
:~\Phi_(x) = x^ + x^ + x^ + x^5 + 1
:~\Phi_(x) = x^ - x^ + x^ - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
:~\Phi_(x) = x^ + x^9 + 1
:~\Phi_(x) = x^ - x^ + x^8 - x^6 + x^4 - x^2 + 1
:~\Phi_(x) = x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^ + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1
:~\Phi_(x) = x^8 + x^7 - x^5 - x^4 - x^3 + x + 1
The case of the 105th cyclotomic polynomial is interesting because 105 is the lowest integer that is the product of three distinct odd prime numbers and this polynomial is the first one that has a coefficient greater than 1:
:\begin
\Phi_(x) = & \; x^ + x^ + x^ - x^ - x^ - 2 x^ - x^ - x^ + x^ + x^ + x^ \\
& + x^ - x^ - x^ - x^ - x^ - x^ + x^ + x^ + x^ \\
& + x^ - x^9 - x^8 - 2 x^7 - x^6 - x^5 + x^2 + x + 1
\end

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアで「Cyclotomic polynomial」の詳細全文を読む



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