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Topic: Root of unity

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	PlanetMath: root of unity
		The roots of unity in any field have many special relationships to one another, some of which are true in general and some of which depend on the field.
		For example, in the ring of endomorphisms of a vector space, the unipotent linear transformations are the closest analogue to roots of unity.
		This is version 11 of root of unity, born on 2001-10-18, modified 2007-07-10.
	planetmath.org /encyclopedia/RootOfUnity.html (343 words)

	Root of unity: Just the facts... (Site not responding. Last check: 2007-10-08)
		The n-th roots of unity form a cyclic group (additional info and facts about cyclic group) of order ((biology) taxonomic group containing one or more families) n under multiplication with 1 as the identity element (An operator that leaves unchanged the element on which it operates).
		The primitive n-th roots of unity are precisely the numbers of the form exp(2πi k/n) where k and n are coprime (additional info and facts about coprime).
		Every nth root of unity is a primitive dth root of unity for exactly one positive divisor (The number by which a dividend is divided) d of n.
	www.absoluteastronomy.com /encyclopedia/r/ro/root_of_unity.htm (763 words)

	Encyclopedia: Root of unity (Site not responding. Last check: 2007-10-08)
		roots of unity or de Moivre numbers are all the complex numbers which yield 1 when raised to a given power n.
		The primitive n -th roots of unity are precisely the numbers of the form exp(2π i k / n) where k and n are coprime.
		Every n -th root of unity is a primitive d -th root of unity for exactly one positive divisor d of n.
	www.nationmaster.com /encyclopedia/Root-of-unity (446 words)

	Welcome to CUTS-International - Consumer Unity & Trust Society
		CUTS International (Consumer Unity and Trust Society) began its journey in 1983 in Rajasthan, from a rural development communication initiative, a wall newspaper Gram Gadar (Village Revolution).
		CUTS Centre for International Trade, Economics and Environment, established in 1996, aims to be a high-level global standard institution for research and advocacy on multilateral trade and sustainable development issues.
		It is being hosted by the Consumer Unity and Trust Society (CUTS), a non-profit making and a leading civil society organization dedicated to research and advocacy on basic economic issues aimed towards economic development and consumer welfare, for more than two decades.
	www.cuts-international.org (822 words)

	Talk:Root of unity - Wikipedia, the free encyclopedia
		Proof #2: the roots of unity are eigenvectors of the discretized Laplacian with periodic boundary conditions, which is Hermitian and therefore has orthogonal eigenvectors.
		The existence of the various proofs of the orthogonality of the roots of unity is indeed important, because it means that this orthogonality appears in different ways in many branches of mathematics.
		The elements of the matrix are not the roots of unity.
	en.wikipedia.org /wiki/Talk:Root_of_unity (2557 words)

	Reference.com/Encyclopedia/Cube root
		In mathematics, the cube root (∛) of a number is a number which, when cubed (multiplied by itself and then multiplied by itself again), gives back the original number.
		The cube root operation is associative with exponentiation and distributive with multiplication and division, but not addition and subtraction.
		If R is one cube root of any real or complex number, the other two cube roots can be found by multiplying R by the two complex cube roots of unity.
	www.reference.com /browse/wiki/Cube_root (311 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Now, some of these are not "primitive" n-th roots (something like that), for example, the 2nd 6th root of unity is also a 3rd root of unity, and as such, is a solution to z^3 - 1 = 0.
		Here, z-1 is the minimal polynomial of the 1st root of unity (1), z+1 is the minimal polynomial for the primitive 2nd root of unity (-1), and z^2+z+1 is the minimal polynomial for the primitive 3rd roots of unity (-1/2 +/- i*sqrt(3)/2).
		Anyway, the minimal polynomial of a primitive n-th root of unity has degree Phi(n), which we wish to be a power of 2.
	www.math.niu.edu /~rusin/papers/known-math/99/constructible (765 words)

	Zach's MathMagic Land
		The n, all distinct, n-th roots of unity are cos (2kpi/n) + i sin (2kpi/n), k= 0, 1,...
		This means that all the roots of unity are points on the unit circle of the complex plane.
		For example, 1 is the primitive first root of unity, -1 is the primitive second root of unity, sqrt(2)/2 + i sqrt(2)/2 and - sqrt(2)/2 - i sqrt(2)/2 are primitive third roots of unity, while i and -i are primitive fourth roots of unity.
	www.math.psu.edu /tseng/class/Math140H/MathMagic.html (1099 words)

	There are trisectable angles that are not constructible
		Taking v to be a primitive 42nd root of unity (angle pi/21), the field generated by it has degree 12 over the rationals, while its cube root (being a primitive 126th root of unity) has degree 36 over the rationals.
		This is nearly a smallest example, since you need at least 7 to prevent constructibility and a factor of 3 to have the cube root pick up a factor of three in its degree over the intermediate field.
		It would seem that the lowest degree solution is given by a 21st root of unity (2pi/21), but this does seem much more succinct when expressed as an angle (vs. pi/21).
	www.cut-the-knot.org /do_you_know/trisect1.shtml (487 words)

	[No title]
		An element of order q^n will be a q^nth root of unity which is not also a q^(n-1)th root of unity (notice that the order of any q^nth root of unity has to be a divisor of q^n and so a power of q).
		Thus, if we can show that there are _exactly_ q^n q^nth roots of unity mod p, it follows that there are at least q^n-q^(n-1) elements of order q^n, because there can be no more than q^(n-1) q^(n-1)th roots of unity.
		Consider the dth roots of unity where d is any factor of p-1.
	math.boisestate.edu /~holmes/holmes/cryptofiles/Notes2.txt (2106 words)

	[No title]
		3) a/(a-1) is also the real part of a root of unity I don't think so; the problem is that the minimal polynomials of the expressions a/(a-1) 'look different' from the minimal polynomials of the expressions a.
		Suppose a is the real part of a primitive n-th root of unity, w.
		Comparing the last two paragraphs we see that the only occasions under which both a and b can be real parts of roots of unity are a few with d (and e) really small.
	www.math.niu.edu /~rusin/known-math/01_incoming/rts_unity (775 words)

	Root Systems
		However, if we choose a representative r_i for each conjugacy class of reflections in the reflection group G and if we choose a root a_i of r_i, then the union of the orbits of the a_i form a suitable set on which G acts as a group of permutations.
		The (i, j)-th entry of the root system matrix for the roots a_1, a_2,..., a_k is delta_(ij) + (alpha_j - 1)(a_i, a_j), where alpha_j is an m-th root of unity, for some m.
		The (i, i)-th entry of the root system matrix is alpha_i and if this is -1, the node is shown as a circle, otherwise it is represented by alpha_i itself.
	www.umich.edu /~gpcc/scs/magma/text514.htm (828 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-08)
		So the computation of the 17th root of unity is extremely important in devising an actual method of construction, for example.
		Conversely, if an n(th) root of unity is expressible using only these operations, a regular polygon of n sides is constructible (so from the value I gave for z[3], the equilateral triangle is constructible).
		The reason that we look at the roots of unity is that mathematicians desire to know the solution sets to equations.
	mathforum.org /library/drmath/view/52231.html (657 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		One answer is "any complex number that when taken to the 'n'th power gives 1." * Example: 1 is a 'n'th root of unity for any n.
		In fact, one can even define e^A when A is a matrix.) - Book terminology: w_n is called the "principal" 'n'th root of unity, defined as the root with the smallest argument (angle in the complex plane): exp(i * 2*pi/n), and all others are defined as powers of it.
		However, as noted in lecture, in general many 'n'th roots of unity satisfy the property that all of the other 'n'th roots are powers of it.
	www.cs.berkeley.edu /~chrishtr/teaching/fall1999-cs170/section-notes-10-13-1999.txt (592 words)

	[ref] 18 Cyclotomic Numbers
		Cyclotomics are usually entered as sums of roots of unity, with rational coefficients, and irrational cyclotomics are displayed in the same way.
		For example, square roots of integers are cyclotomic integers (see ATLAS irrationalities), any root of unity is a cyclotomic integer, character values are always cyclotomic integers, but all rationals which are not integers are not cyclotomic integers.
		A special integral basis of cyclotomic fields is chosen that allows one to easily convert arbitrary sums of roots of unity into the basis, as well as to convert a cyclotomic represented w.r.t.
	www.gap-system.org /Manuals/doc/htm/ref/CHAP018.htm (2212 words)

	When does n \| (2^m -1) for arbitrary odd n? \| Ask MetaFilter (Site not responding. Last check: 2007-10-08)
		I'm reading a paper in which the author assumes the existance of "a primitive nth root of unity in GF(2^m) that is an appropriate extension of GF(2)" for an arbitrary odd number, n.
		Example: roots of unity in the complex numbers.
		Proof: a finite abelian group is cyclic iff for all n, there are at most n elements whose nth power is the identity, which is easily seen from the classification of finite abelian groups.
	ask.metafilter.com /mefi/24349 (702 words)

	Creation Functions
		Given a positive integer m > 2, create the field obtained by adjoining the m-th roots of unity to Q. It is possible to assign a name to the primitive m-th root of unity zeta_(m) using angle brackets: R ~~:= CyclotomicField(m).~~
		Primitive roots of unity zeta_m are chosen in such a way that zeta_m^(m/d)=zeta_d, for every divisor d of m; one may think of this as choosing zeta_m=(e)^(2pi i/m) in the complex plane for every m (a convention that is followed for the explicit embedding in the complex domains).
		Given a cyclotomic field Q(zeta_m) and an integer n>2, create the n-th root of unity zeta_n in K. An error results if zeta_n notin K, that is, if n does not divide m (or 2m in case m is odd).
	www.math.uiuc.edu /Software/magma/text350.html (523 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Let K = Q(; ;* * j), where is a primitive p-th root of unity, is a primitive (p - 1)-st root of un* ity, and j is a (p - 1)-st root* of -p.
		In the above notation a cube root of unity z 2 D is given by t* *he formula 1 p___(i + 1) 1 i z = -__+ -2 ______ S = -__+ __S: 2 4 2 2 Proof.
		Since Q3 does not contain a primitive cube root of unity, the norm of z is eq* *ual to 1 and we conclude that z 2 Sl.
	hopf.math.purdue.edu /Gorbounov-Mahowald-Symonds/peterme.txt (4341 words)

	Factoring
		st roots of unity, and these are all the nonzero elements of the field.
		roots of unity must exist in some field.
		If we list the exponents of the primitive roots of unity, we get what are called the cylotomic cosets.
	www.engineering.usu.edu /classes/ece/7670/lecture4/node3.html (181 words)

	ipedia.com: Root of unity Article (Site not responding. Last check: 2007-10-08)
		In mathematics, the n -th roots of unity or de Moivre Numbers, named after Abraham de Moivre, are complex numbers located on the unit circle.
		In mathematics, the n-th roots of unity or de Moivre Numbers, named after Abraham de Moivre, are complex numbers located on the unit circle.
		Conversely, every abelian extension of the rationals is such a subfield of a cyclotomic field - a theorem of Kronecker, usually called the Kronecker-Weber theorem on the grounds that Weber supplied the proof.
	www.ipedia.com /root_of_unity.html (577 words)

	square root on Encyclopedia.com (Site not responding. Last check: 2007-10-08)
		Root segregation of C3 and C4 species using carbon isotope composition.(NOTES)
		Death of high Modernism; Escaping from the grey: Regionalism and the pursuit of roots; High-Tech; PoMo and neo Neo-Classicism.
		Square Roots owner Jennifer Newman greets customers with fresh flowers and folk art.
	www.encyclopedia.com /html/X/X-squarero.asp (507 words)

	Citebase - Relativistic Toda chain at root of unity III. Relativistic Toda chain hierarchy (Site not responding. Last check: 2007-10-08)
		Relativistic Toda chain at root of unity III.
		Relativistic Toda chain at root of unity II.
		Matrix elements of quantum intertwiner as well as the modified Q-operator for the quantum relativistic Toda chain at root of unity are constructed explicitly.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:nlin/0107063 (929 words)

	CS 480/680-504 -- Final Exam (Site not responding. Last check: 2007-10-08)
		Determine a 8th and 256-th root of unity in Z_769.
		Write a Maple procedure that constructs the inverse of the n-point DFT matrix with elements mod p (you should provide a primitive n-th root of unity as an argument).
		You must provide p as a parameter in addition to the size n, two inputs, and a primitive n-th root of unity.
	www.mcs.drexel.edu /~jjohnson/fa01/asym/assignments/final.html (366 words)

	Garvey’s vision of unity takes root (Site not responding. Last check: 2007-10-08)
		On African Liberation Day at the African Orthodox Church (which was founded by Marcus Garvey in 1921 in Harlem), members of the UNIA, African Communities League, Nation of Islam, New Black Panther Party and representatives of African countries gathered to announce the unity of Africans throughout the Diaspora.
		He also stressed the value of unity in dealing with the condition of our communities.
		We want millions of more men, women and children to come to D.C. on October 14th, 15th and 16th, in the spirit of Marcus Mosiah Garvey,” he said.
	www.finalcall.com /artman/publish/article_2056.shtml (514 words)

	Creation Functions
		Cyclotomic fields can be created from an integer specifying which roots of unity it should contain or from a collection of elements of an existing field or order.
		Given a positive integer m, create the field obtained by adjoining the m-th roots of unity to Q. It is possible to assign a name to the primitive m-th root of unity zeta_(m) using angle brackets:
		Given a cyclotomic field K = Q(zeta_m) and an integer n>2, create the n-th root of unity zeta_n in K. An error results if zeta_n notin K, that is, if n does not divide m (or 2m in case m is odd).
	www.umich.edu /~gpcc/scs/magma/text654.htm (586 words)

	root of unity :: Mathematical Dictionary, VeryPrime.com (Site not responding. Last check: 2007-10-08)
		If n `>=` 1 is an integer, then an nth root of unity is a complex number `zeta ` with `zeta^n` = 1.
		`zeta^n` = 1 are `(1, ntheta)` = (1, 0), because the nth roots of unity are equally spaced around the unit circle.
		Marian Olejar, Jr.: root of unity from VeryPrime's Dictionary of mathematics
	www.veryprime.com /dict/root_of_unity.php (144 words)

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