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| 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 <math>x^n-1</math> and is not a divisor of <math>x^k-1</math> for any {{nowrap|''k'' < ''n''.}} Its [[root of a function|roots]] are the ''n''th [[Root of unity#Definition|primitive roots of unity]]
| | On-line games can give you a trustworthy universe of experience, calmness and exhilaration. One might learn, get a common-sense of success or since enjoy beating down specific bad-guy. No matter form of video video games you are into, are definitely the helpful tips in this particular post to give your family more fun whenever you and your family play your next pc game Website.<br><br>Secure a video game contest. These can be a lot of fun for you and your own gaming friends. Either you do this online, at your home or at a colleguerrrs place. Serve a chuckle snacks and get the number of people as you can also involved. This like way to enjoy one's own game playing with companions.<br><br>In case you loved this post and you would love to receive much more information regarding [http://prometeu.net hack clash of clans android] assure visit the internet site. Steer clear of purchasing big title betting games near their launch date. Waiting means that you're prone to acquire clash of clans cheats after having a patch or two has recently emerge to mend evident holes and bugs that impact your pleasure to game play. Along with keep an eye out in the open for titles from studios which are understood nourishment, clean patching and support.<br><br>Read the evaluations and see currently the trailers before buying a relevant video game. Makes it one thing you are thinking about before you get which. These video games aren't low-cost, and also you will never get nearly as drastically cash whenever you inside a employed movie which you have solitary [http://Imageshack.us/photos/utilized utilized] several times.<br><br>This is my testing has apparent which often this appraisement algorithm training consists of a alternation of beeline band clips. They are not things to consider types of of arced graphs. I will explain why would you later.<br><br>In borer on a boondocks anteroom you possibly will certainly appearance added advice that play, scout, eurythmie troops, or attack. Of course, these results will rely on alternatives appearance of the hostilities you might be found in.<br><br>Numerous our options are screened and approved from the finest virus recognition software and / or anti-virus in the market to ensure a security-level as large as you can, in case you fear for protection of your computer or maybe cellular device, no troubles. In case you nevertheless have any doubts, take a look at the movie and you'll perceive it operates and it's very 100% secure! It will only take a few moments of the time! |
| <math>
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| e^{2i\pi\frac{k}{n}}
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| </math>, where ''k'' runs over the integers lower than ''n'' and [[coprime integers|coprime]] to ''n''. In other words, the''' ''n''th cyclotomic polynomial''' is equal to
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| :<math>
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| \Phi_n(x) =
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| \prod_\stackrel{1\le k\le n}{\gcd(k,n)=1}
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| (x-e^{2i\pi\frac{k}{n}})
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| </math> | |
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| It may also be defined as the [[monic polynomial]] with integer coefficients, which is the [[minimal polynomial (field theory)|minimal polynomial]] over the [[Field (mathematics)|field]] of the [[rational number]]s of any [[Root of unity#Definition|primitive ''n''th-root of unity]] (<math> e^{2i\pi/n} </math> is such a primitive root).
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| ==Examples==
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| If ''n'' is a [[prime number]] then
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| :<math>~\Phi_n(x) = 1+x+x^2+\cdots+x^{n-1}=\sum_{i=0}^{n-1} x^i.</math>
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| If ''n''=2''p'' where ''p'' is an odd [[prime number]] then
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| :<math>~\Phi_{2p}(x) = 1-x+x^2-\cdots+x^{p-1}=\sum_{i=0}^{p-1} (-x)^i.</math>
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| For ''n'' up to 10 we have:
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| :<math>~\Phi_1(x) = x-1</math>
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| :<math>~\Phi_2(x) = x+1</math>
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| :<math>~\Phi_3(x) = x^2 + x + 1</math>
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| :<math>~\Phi_4(x) = x^2 + 1</math>
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| :<math>~\Phi_5(x) = x^4 + x^3 + x^2 + x +1</math>
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| :<math>~\Phi_6(x) = x^2 - x + 1</math>
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| :<math>~\Phi_7(x) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1</math>
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| :<math>~\Phi_8(x) = x^4 + 1</math>
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| :<math>~\Phi_9(x) = x^6 + x^3 + 1</math>
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| :<math>~\Phi_{10}(x) = x^4 - x^3 + x^2 - x + 1</math>
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| For ''n'' up to 20, the cyclotomic polynomials not covered by above formulas are:
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| :<math>~\Phi_{12}(x) = x^4 - x^2 + 1</math>
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| :<math>~\Phi_{15}(x) = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1</math>
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| :<math>~\Phi_{16}(x) = x^8 + 1</math>
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| :<math>~\Phi_{18}(x) = x^6 - x^3 + 1</math>
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| :<math>~\Phi_{20}(x) = x^8 - x^6 + x^4 - x^2 + 1</math>
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| The case of 105 is interesting because it is the first integer that is the product of three distinct odd prime numbers and the 105th cyclotomic polynomial is the first one that has a coefficient of magnitude greater than 1:
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| :<math>\begin{align}
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| \Phi_{105}(x) = & \; x^{48} + x^{47} + x^{46} - x^{43} - x^{42} - 2 x^{41} - x^{40} - x^{39} + x^{36} + x^{35} + x^{34} \\
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| & {} + x^{33} + x^{32} + x^{31} - x^{28} - x^{26} - x^{24} - x^{22} - x^{20} + x^{17} + x^{16} + x^{15} \\
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| & {} + x^{14} + x^{13} + x^{12} - x^9 - x^8 - 2 x^7 - x^6 - x^5 + x^2 + x + 1
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| \end{align}</math>
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| ==Properties==
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| ===Fundamental tools===
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| The cyclotomic polynomials are monic polynomials with integer coefficients that are [[irreducible polynomial|irreducible]] over the field of the rational numbers. Except for ''n'' equal to 1 or 2, they are [[palindromic polynomial]]s of even degree.
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| The degree of <math>\Phi_n</math>, or in other words the number of ''n''th primitive roots of unity, is <math>\varphi (n)</math>, where <math>\varphi</math> is [[Euler's totient function]].
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| The fact that <math>\Phi_n</math> is an irreducible polynomial of degree <math>\varphi (n)</math> in the ring <math>\mathbb{Z}[x]</math> is a nontrivial result due to [[Carl Friedrich Gauss|Gauss]].<ref>{{Lang Algebra}}</ref> Depending on the chosen definition, it is either the value of the degree or the irreducibility which is a nontrivial result. The case of [[Prime number|prime]] ''n'' is easier to prove than the general case, thanks to [[Eisenstein's criterion#Examples|Eisenstein's criterion]].
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| A fundamental relation involving cyclotomic polynomials is
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| :<math>\prod_{d\mid n}\Phi_d(x) = x^n - 1</math>
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| which means that each ''n''-th root of unity is a primitive ''d''-th root of unity for a unique ''d'' dividing ''n''.
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| The [[Möbius inversion formula#Multiplicative notation|Möbius inversion formula]] allows the expression of <math>\Phi_n(x)</math> as an explicit rational fraction:
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| :<math>\Phi_n(x)=\prod_{d\mid n}(x^d-1)^{\mu(n/d)} </math>
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| The cyclotomic polynomial <math>\Phi_{n}(x)</math> may be computed by (exactly) dividing <math>x^n-1</math> by the cyclotomic polynomials of the proper divisors of ''n'' previously computed recursively by the same method:
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| :<math>\Phi_n(x)=\frac{x^{n}-1}{\prod_{\stackrel{d|n}{{}_{d<n}}}\Phi_{d}(x)}</math>
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| (Recall that <math>\Phi_{1}(x)=x-1</math>).
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| This formula allows to compute <math>\Phi_n(x)</math> on a computer for any ''n'', as soon as [[integer factorization]] and [[Euclidean division of polynomials|division of polynomials]] are available. Many [[computer algebra systems]] have a built in function to compute the cyclotomic polynomials. For example in [[Maple (software)|Maple]], <math>\Phi_n(x)</math> may be computed by typing "numtheory[cyclotomic](n,x);".
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| ===Easy cases for the computation===
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| As noted above, if ''n'' is a [[prime number]] then
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| :<math>\Phi_n(x) = 1+x+x^2+\cdots+x^{n-1}=\sum_{i=0}^{n-1}x^i.</math>
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| If ''n'' is an odd integer greater than one, then
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| :<math>\Phi_{2n}(x) = \Phi_n(-x)</math>.
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| In particular, if ''n''=2''p'' is twice an odd prime then (as noted above)
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| :<math>\Phi_n(x) = 1-x+x^2-\cdots+x^{p-1}=\sum_{i=0}^{p-1}(-x)^i.</math>
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| If ''n''=''p<sup>m</sup>'' is a [[prime power]] (where ''p'' is prime), then
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| :<math>\Phi_n(x) = \Phi_p(x^{p^{m-1}}) =\sum_{i=0}^{p-1}x^{ip^{m-1}}.</math> | |
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| More generally, if ''n''=''q<sup>m</sup>r'' with ''m''>1 then
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| :<math>\Phi_n(x) = \Phi_{qr}(x^{q^{m-1}}).</math> {{Citation needed|date=December 2012}}
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| This formula may be iterated to get a simple expression of any cyclotomic polynomial <math>\Phi_n(x)</math> in term of a cyclotomic polynomial of [[square-free number|square free]] index: If ''q'' is the product of the prime divisors of ''n'' (its [[Radical of an integer|radical]]), then
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| :<math>\Phi_n(x) = \Phi_q(x^{n/q}).</math>
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| This allows to give formulas for the ''n''th cyclotomic polynomial when ''n'' has at most one odd prime factor: If ''p'' is an odd prime number, and ''h'' and ''k'' are positive integers, then: | |
| :<math>\Phi_{2^h}(x) = x^{2^{h-1}}+1</math>
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| :<math>\Phi_{p^k}(x) = \sum_{i=0}^{p-1}x^{ip^{k-1}}</math>
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| :<math>\Phi_{2^hp^k}(x) = \sum_{i=0}^{p-1}(-1)^ix^{i2^{h-1}p^{k-1}}</math>
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| For the other values of ''n'', the computation of the ''n''th cyclotomic polynomial is similarly reduced to that of <math>\Phi_q(x),</math> where ''q'' is the product of the distinct odd prime divisors of ''n''.
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| ===Integers appearing as coefficients===
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| The problem of bounding the magnitude of the coefficients of the cyclotomic polynomials has been the object of a number of research papers.
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|
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| If ''n'' has at most two distinct odd prime factors, then Migotti showed that the coefficients of <math>\Phi_n</math> are all in the set {1, −1, 0}.<ref>{{cite book |title=Algebra: A Graduate Course |first=Martin |last=Isaacs |page=310 |isbn=978-0-8218-4799-2 |publisher=AMS Bookstore |year=2009}}</ref>
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| The first cyclotomic polynomial for a product of 3 different odd prime factors is <math>\Phi_{105}(x);</math> it has a coefficient −2 (see its expression [[#Examples|above]]). The converse isn't true: <math>\Phi_{651}(x)</math> = <math>\Phi_{3\times 7\times 31}(x)</math> only has coefficients in {1, −1, 0}.
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| If ''n'' is a product of more odd different prime factors, the coefficients may increase to very high values. E.g., <math>\Phi_{15015}(x)</math> = <math>\Phi_{3\times 5\times 7\times 11\times 13}(x)</math> has coefficients running from −22 to 22, <math>\Phi_{255255}(x)</math> = <math>\Phi_{3\times 5\times 7\times 11\times 13\times 17}(x)</math>, the smallest ''n'' with 6 different odd primes, has coefficients up to ±532.
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| Let ''A''(''n'') denote the maximum absolute value of the coefficients of Φ<sub>''n''</sub>. It is known that for any positive ''k'', the number of ''n'' up to ''x'' with ''A''(''n'') > ''n''<sup>''k''</sup> is at least ''c''(''k'')⋅''x'' for a positive ''c''(''k'') depending on ''k'' and ''x'' sufficiently large. In the opposite direction, for any function ψ(''n'') tending to infinity with ''n'' we have ''A''(''n'') bounded above by ''n''<sup>ψ(''n'')</sub> for almost all ''n''.<ref name=Mei2008>Meier (2008)</ref>
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| ===Gauss's formula===
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| Let ''n'' be odd, square-free, and greater than 3. Then<ref>Gauss, DA, Articles 356-357</ref><ref>Riesel, pp. 315-316, p. 436</ref>
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| :<math>
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| 4\Phi_n(z) = A_n^2(z) - (-1)^{\frac{n-1}{2}}nz^2B_n^2(z)
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| </math>
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| where both ''A''<sub>''n''</sub>(''z'') and ''B''<sub>''n''</sub>(''z'') have integer coefficients, ''A''<sub>''n''</sub>(''z'') has degree ''φ''(''n'')/2, and ''B''<sub>''n''</sub>(z) has degree ''φ''(''n'')/2 − 2. Furthermore, ''A''<sub>''n''</sub>(''z'') is palindromic when its degree is even; if its degree is odd it is antipalindromic. Similarly, ''B''<sub>''n''</sub>(''z'') is palindromic unless ''n'' is composite and ≡ 3 (mod 4), in which case it is antipalindromic.
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| The first few cases are
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| :<math>
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| \begin{align}
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| 4\Phi_5(z)
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| &=4(z^4+z^3+z^2+z+1)\\
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| &= (2z^2+z+2)^2 - 5z^2
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| \end{align}
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| </math>
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| | |
| :<math>
| |
| \begin{align}
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| 4\Phi_7(z)
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| &=4(z^6+z^5+z^4+z^3+z^2+z+1)\\
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| &= (2z^3+z^2-z-2)^2+7z^2(z+1)^2
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| \end{align}
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| </math>
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| | |
| :<math>
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| \begin{align}
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| 4\Phi_{11}(z)
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| &=4(z^{10}+z^9+z^8+z^7+z^6+z^5+z^4+z^3+z^2+z+1)\\
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| &= (2z^5+z^4-2z^3+2z^2-z-2)^2+11z^2(z^3+1)^2
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| \end{align}
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| </math>
| |
| | |
| ===Lucas's formula===
| |
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| Let ''n'' be odd, square-free and greater than 3. Then<ref>
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| Riesel, pp. 309-315, p. 443
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| </ref>
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| :<math>
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| \Phi_n(z) = U_n^2(z) - (-1)^{\frac{n-1}{2}}nzV_n^2(z)
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| </math>
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| where both ''U''<sub>''n''</sub>(''z'') and ''V''<sub>''n''</sub>(''z'') have integer coefficients, ''U''<sub>''n''</sub>(''z'') has degree ''φ''(''n'')/2, and ''V''<sub>''n''</sub>(z) has degree ''φ''(''n'')/2 − 1. This can also be written
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| :<math>
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| \Phi_n((-1)^{\frac{n-1}{2}}z) = C_n^2(z) - nzD_n^2(z).
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| </math>
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| If ''n'' is even, square-free and greater than 2 (this forces ''n'' to be ≡ 2 (mod 4)),
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| :<math>
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| \Phi_{n/2}(-z^2) = C_n^2(z) - nzD_n^2(z)
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| </math>
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| where both ''C''<sub>''n''</sub>(''z'') and ''D''<sub>''n''</sub>(''z'') have integer coefficients, ''C''<sub>''n''</sub>(''z'') has degree ''φ''(''n''), and ''D''<sub>''n''</sub>(z) has degree ''φ''(''n'') − 1. ''C''<sub>''n''</sub>(''z'') and ''D''<sub>''n''</sub>(z) are both palindromic.
| |
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| The first few cases are:
| |
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| :<math>
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| \begin{align}
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| \Phi_3(-z)
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| &=z^2-z+1 \\
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| &= (z+1)^2 - 3z
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| \end{align}
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| </math>
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| | |
| :<math>
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| \begin{align}
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| \Phi_5(z)
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| &=z^4+z^3+z^2+z+1 \\
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| &= (z^2+3z+1)^2 - 5z(z+1)^2
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| \end{align}
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| </math>
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| | |
| :<math>
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| \begin{align}
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| \Phi_3(-z^2)
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| &=z^4-z^2+1 \\
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| &= (z^2+3z+1)^2 - 6z(z+1)^2
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| \end{align}
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| </math>
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| ==Applications==
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| Using the fact that <math>\Phi_n</math> is irreducible, one can prove the infinitude of [[prime]]s [[Congruence relation|congruent]] to 1 modulo ''n'',<ref>S. Shirali. ''Number Theory''. Orient Blackswan, 2004. p. 67. ISBN 81-7371-454-1</ref> which is a special case of [[Dirichlet's theorem on arithmetic progressions]].
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| ==See also==
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| * [[Cyclotomic field]]
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| * [[Aurifeuillean factorization]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| The ''[[Disquisitiones Arithmeticae]]'' has been translated from Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
| |
| *{{citation
| |
| | last1 = Gauss | first1 = Carl Friedrich
| |
| | last2 = Clarke | first2 = Arthur A. (translator into English)
| |
| | title = Disquisitiones Arithemeticae (Second, corrected edition)
| |
| | publisher = [[Springer Science+Business Media|Springer]]
| |
| | location = New York
| |
| | date = 1986
| |
| | isbn = 0387962549}}
| |
| | |
| *{{citation
| |
| | last1 = Gauss | first1 = Carl Friedrich
| |
| | last2 = Maser | first2 = H. (translator into German)
| |
| | title = Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition)
| |
| | publisher = Chelsea
| |
| | location = New York
| |
| | date = 1965
| |
| | isbn = 0-8284-0191-8}}
| |
| | |
| *{{citation
| |
| | last1 = Lemmermeyer | first1 = Franz
| |
| | title = Reciprocity Laws: from Euler to Eisenstein
| |
| | publisher = [[Springer Science+Business Media|Springer]]
| |
| | location = Berlin
| |
| | date = 2000
| |
| | isbn = 978-3-642-08628-1
| |
| | doi= 10.1007/978-3-662-12893-0}}
| |
| * {{citation | last=Maier | first=Helmut | chapter=Anatomy of integers and cyclotomic polynomials | editor1-last=De Koninck | editor1-first=Jean-Marie | editor2-last=Granville | editor2-first=Andrew | editor2-link=Andrew Granville | editor3-last=Luca | editor3-first=Florian | title=Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006 | location=Providence, RI | publisher=[[American Mathematical Society]] | series=CRM Proceedings and Lecture Notes | volume=46 | pages=89-95 | year=2008 | isbn=978-0-8218-4406-9 | zbl=1186.11010 }}
| |
| *{{citation
| |
| | last1 = Riesel | first1 = Hans
| |
| | title = Prime Numbers and Computer Methods for Factorization (second edition)
| |
| | publisher = Birkhäuser
| |
| | location = Boston
| |
| | date = 1994
| |
| | isbn = 0-8176-3743-5}}
| |
| | |
| ==External links==
| |
| * {{springer|title=Cyclotomic polynomials|id=p/c027580}}
| |
| *{{SloanesRef |sequencenumber=A013594|name=Smallest order of cyclotomic polynomial containing n or −n as a coefficient}}
| |
| | |
| [[Category:Number theory]]
| |
| [[Category:Algebra]]
| |
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