|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| 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]]
| | We all like to reduce our fat swiftly. We wish To look sleek plus fit for summer activities and social outings. The toughest part of any fat reduction system, nevertheless, is usually simply getting the motivation to begin, and building a strong foundation for success. Because you need to reach a goals, remember that the greatest destination to begin is the beginning.<br><br>Food is the fuel of the body plus you want to recognize which though fasting is included inside the long list of techniques to lose weight it is very at the finish of the day a very unhealthy practice. Instead of having oily and spiced up food stuffs you need to commence the usage of cereals and foods that have a higher fiber plus grain content. One of the better methods to lose weight fast is the intake of 2 servings of fruit plus 5 servings of vegetables a day.<br><br>To lose weight too swiftly might put a strain on your program and may aggravate some of the underlying illnesses you have. You have to lose the weight the same technique we gained it: One pound at a time.<br><br>Eat plenty of raw veggies. Start a meal with many salad having carrots, peppers, celery, broccoli cucumber plus cabbage. Eating raw greens usually give we a feeling of fullness, causing we to consume less at standard food.<br><br>11 Say no to white carbohydrates. This includes white rice, white flour products like bread, pastries and starchy veggies including [http://safedietplansforwomen.com/how-to-lose-weight-fast lose weight] potatoes. Instead loads up your body on fiber rich foods and lean protein.<br><br>My first ever lengthy haul flight to Toronto, I did not have adequate rest for more than 20 hours including flight time. (With because of the stinky heavy snoring man whom sat alongside me). In my luggage, I absolutely brought my workout attire plus I was intending to create full employ of the resort's fitness facilities. However the time I managed to get inside my hotel space, I found myself so depleted which I really changed into anything comfortable, turned off the lights plus crashed - for 14 hours directly. I never had the strength to use the wellness club inside any way throughout my four day remain .<br><br>Jogging, when done correctly, may confirm to be a whimsical method of losing fat at the fastest pace. But, it happens to be important to keep inside mind all the factors and techniques before taking up jogging. A consistent routine plus adherence to all the pointers can keep you healthy and fit. |
| <math>
| |
| e^{2i\pi\frac{k}{n}}
| |
| </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
| |
| :<math>
| |
| \Phi_n(x) =
| |
| \prod_\stackrel{1\le k\le n}{\gcd(k,n)=1}
| |
| (x-e^{2i\pi\frac{k}{n}})
| |
| </math> | |
| | |
| 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).
| |
| | |
| ==Examples==
| |
| | |
| If ''n'' is a [[prime number]] then
| |
| :<math>~\Phi_n(x) = 1+x+x^2+\cdots+x^{n-1}=\sum_{i=0}^{n-1} x^i.</math>
| |
| If ''n''=2''p'' where ''p'' is an odd [[prime number]] then
| |
| :<math>~\Phi_{2p}(x) = 1-x+x^2-\cdots+x^{p-1}=\sum_{i=0}^{p-1} (-x)^i.</math>
| |
| For ''n'' up to 10 we have:
| |
| :<math>~\Phi_1(x) = x-1</math>
| |
| | |
| :<math>~\Phi_2(x) = x+1</math>
| |
| | |
| :<math>~\Phi_3(x) = x^2 + x + 1</math>
| |
| | |
| :<math>~\Phi_4(x) = x^2 + 1</math>
| |
| | |
| :<math>~\Phi_5(x) = x^4 + x^3 + x^2 + x +1</math>
| |
| | |
| :<math>~\Phi_6(x) = x^2 - x + 1</math>
| |
| | |
| :<math>~\Phi_7(x) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1</math>
| |
| | |
| :<math>~\Phi_8(x) = x^4 + 1</math>
| |
| | |
| :<math>~\Phi_9(x) = x^6 + x^3 + 1</math>
| |
| | |
| :<math>~\Phi_{10}(x) = x^4 - x^3 + x^2 - x + 1</math>
| |
| | |
| For ''n'' up to 20, the cyclotomic polynomials not covered by above formulas are:
| |
| | |
| :<math>~\Phi_{12}(x) = x^4 - x^2 + 1</math>
| |
| :<math>~\Phi_{15}(x) = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1</math>
| |
| :<math>~\Phi_{16}(x) = x^8 + 1</math>
| |
| :<math>~\Phi_{18}(x) = x^6 - x^3 + 1</math>
| |
| :<math>~\Phi_{20}(x) = x^8 - x^6 + x^4 - x^2 + 1</math>
| |
| | |
| 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:
| |
| :<math>\begin{align}
| |
| \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} \\
| |
| & {} + x^{33} + x^{32} + x^{31} - x^{28} - x^{26} - x^{24} - x^{22} - x^{20} + x^{17} + x^{16} + x^{15} \\
| |
| & {} + x^{14} + x^{13} + x^{12} - x^9 - x^8 - 2 x^7 - x^6 - x^5 + x^2 + x + 1
| |
| \end{align}</math>
| |
| | |
| ==Properties==
| |
| | |
| ===Fundamental tools===
| |
| | |
| 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.
| |
| | |
| 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]].
| |
| | |
| 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]].
| |
| | |
| A fundamental relation involving cyclotomic polynomials is
| |
| :<math>\prod_{d\mid n}\Phi_d(x) = x^n - 1</math>
| |
| which means that each ''n''-th root of unity is a primitive ''d''-th root of unity for a unique ''d'' dividing ''n''.
| |
| | |
| The [[Möbius inversion formula#Multiplicative notation|Möbius inversion formula]] allows the expression of <math>\Phi_n(x)</math> as an explicit rational fraction:
| |
| :<math>\Phi_n(x)=\prod_{d\mid n}(x^d-1)^{\mu(n/d)} </math>
| |
| | |
| 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:
| |
| | |
| :<math>\Phi_n(x)=\frac{x^{n}-1}{\prod_{\stackrel{d|n}{{}_{d<n}}}\Phi_{d}(x)}</math>
| |
| | |
| (Recall that <math>\Phi_{1}(x)=x-1</math>).
| |
| | |
| 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);".
| |
| | |
| ===Easy cases for the computation===
| |
| As noted above, if ''n'' is a [[prime number]] then
| |
| :<math>\Phi_n(x) = 1+x+x^2+\cdots+x^{n-1}=\sum_{i=0}^{n-1}x^i.</math>
| |
| | |
| If ''n'' is an odd integer greater than one, then
| |
| :<math>\Phi_{2n}(x) = \Phi_n(-x)</math>.
| |
| | |
| In particular, if ''n''=2''p'' is twice an odd prime then (as noted above)
| |
| :<math>\Phi_n(x) = 1-x+x^2-\cdots+x^{p-1}=\sum_{i=0}^{p-1}(-x)^i.</math>
| |
| | |
| If ''n''=''p<sup>m</sup>'' is a [[prime power]] (where ''p'' is prime), then
| |
| :<math>\Phi_n(x) = \Phi_p(x^{p^{m-1}}) =\sum_{i=0}^{p-1}x^{ip^{m-1}}.</math>
| |
| | |
| More generally, if ''n''=''q<sup>m</sup>r'' with ''m''>1 then
| |
| :<math>\Phi_n(x) = \Phi_{qr}(x^{q^{m-1}}).</math> {{Citation needed|date=December 2012}}
| |
| | |
| 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 | |
| :<math>\Phi_n(x) = \Phi_q(x^{n/q}).</math>
| |
| | |
| 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>
| |
| :<math>\Phi_{p^k}(x) = \sum_{i=0}^{p-1}x^{ip^{k-1}}</math>
| |
| :<math>\Phi_{2^hp^k}(x) = \sum_{i=0}^{p-1}(-1)^ix^{i2^{h-1}p^{k-1}}</math>
| |
| 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''.
| |
| | |
| ===Integers appearing as coefficients===
| |
| | |
| The problem of bounding the magnitude of the coefficients of the cyclotomic polynomials has been the object of a number of research papers.
| |
|
| |
| 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>
| |
| | |
| 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}.
| |
| | |
| 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.
| |
| | |
| 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>
| |
| | |
| ===Gauss's formula===
| |
| | |
| 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>
| |
| | |
| :<math>
| |
| 4\Phi_n(z) = A_n^2(z) - (-1)^{\frac{n-1}{2}}nz^2B_n^2(z)
| |
| </math>
| |
| | |
| 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.
| |
| | |
| The first few cases are
| |
| | |
| :<math>
| |
| \begin{align}
| |
| 4\Phi_5(z)
| |
| &=4(z^4+z^3+z^2+z+1)\\
| |
| &= (2z^2+z+2)^2 - 5z^2
| |
| \end{align}
| |
| </math>
| |
| | |
| :<math>
| |
| \begin{align}
| |
| 4\Phi_7(z)
| |
| &=4(z^6+z^5+z^4+z^3+z^2+z+1)\\
| |
| &= (2z^3+z^2-z-2)^2+7z^2(z+1)^2
| |
| \end{align}
| |
| </math>
| |
| | |
| :<math>
| |
| \begin{align}
| |
| 4\Phi_{11}(z)
| |
| &=4(z^{10}+z^9+z^8+z^7+z^6+z^5+z^4+z^3+z^2+z+1)\\
| |
| &= (2z^5+z^4-2z^3+2z^2-z-2)^2+11z^2(z^3+1)^2
| |
| \end{align}
| |
| </math>
| |
| | |
| ===Lucas's formula===
| |
| | |
| Let ''n'' be odd, square-free and greater than 3. Then<ref>
| |
| Riesel, pp. 309-315, p. 443
| |
| </ref>
| |
| | |
| :<math>
| |
| \Phi_n(z) = U_n^2(z) - (-1)^{\frac{n-1}{2}}nzV_n^2(z)
| |
| </math>
| |
| | |
| 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
| |
| | |
| :<math>
| |
| \Phi_n((-1)^{\frac{n-1}{2}}z) = C_n^2(z) - nzD_n^2(z).
| |
| </math>
| |
| | |
| If ''n'' is even, square-free and greater than 2 (this forces ''n'' to be ≡ 2 (mod 4)),
| |
| | |
| :<math>
| |
| \Phi_{n/2}(-z^2) = C_n^2(z) - nzD_n^2(z)
| |
| </math>
| |
| | |
| 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.
| |
| | |
| The first few cases are:
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \Phi_3(-z)
| |
| &=z^2-z+1 \\
| |
| &= (z+1)^2 - 3z
| |
| \end{align}
| |
| </math>
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \Phi_5(z)
| |
| &=z^4+z^3+z^2+z+1 \\
| |
| &= (z^2+3z+1)^2 - 5z(z+1)^2
| |
| \end{align}
| |
| </math>
| |
| | |
| :<math>
| |
| \begin{align}
| |
| \Phi_3(-z^2)
| |
| &=z^4-z^2+1 \\
| |
| &= (z^2+3z+1)^2 - 6z(z+1)^2
| |
| \end{align}
| |
| </math>
| |
| | |
| ==Applications==
| |
| | |
| 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]].
| |
| | |
| ==See also==
| |
| * [[Cyclotomic field]]
| |
| * [[Aurifeuillean factorization]]
| |
| | |
| ==Notes==
| |
| | |
| {{Reflist}}
| |
| | |
| ==References==
| |
| | |
| 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]]
| |
We all like to reduce our fat swiftly. We wish To look sleek plus fit for summer activities and social outings. The toughest part of any fat reduction system, nevertheless, is usually simply getting the motivation to begin, and building a strong foundation for success. Because you need to reach a goals, remember that the greatest destination to begin is the beginning.
Food is the fuel of the body plus you want to recognize which though fasting is included inside the long list of techniques to lose weight it is very at the finish of the day a very unhealthy practice. Instead of having oily and spiced up food stuffs you need to commence the usage of cereals and foods that have a higher fiber plus grain content. One of the better methods to lose weight fast is the intake of 2 servings of fruit plus 5 servings of vegetables a day.
To lose weight too swiftly might put a strain on your program and may aggravate some of the underlying illnesses you have. You have to lose the weight the same technique we gained it: One pound at a time.
Eat plenty of raw veggies. Start a meal with many salad having carrots, peppers, celery, broccoli cucumber plus cabbage. Eating raw greens usually give we a feeling of fullness, causing we to consume less at standard food.
11 Say no to white carbohydrates. This includes white rice, white flour products like bread, pastries and starchy veggies including lose weight potatoes. Instead loads up your body on fiber rich foods and lean protein.
My first ever lengthy haul flight to Toronto, I did not have adequate rest for more than 20 hours including flight time. (With because of the stinky heavy snoring man whom sat alongside me). In my luggage, I absolutely brought my workout attire plus I was intending to create full employ of the resort's fitness facilities. However the time I managed to get inside my hotel space, I found myself so depleted which I really changed into anything comfortable, turned off the lights plus crashed - for 14 hours directly. I never had the strength to use the wellness club inside any way throughout my four day remain .
Jogging, when done correctly, may confirm to be a whimsical method of losing fat at the fastest pace. But, it happens to be important to keep inside mind all the factors and techniques before taking up jogging. A consistent routine plus adherence to all the pointers can keep you healthy and fit.