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Topic: Eisenstein integer

	Eisenstein integer - Wikipedia, the free encyclopedia
		The Eisenstein integers form a triangular lattice in the complex plane.
		The group of units in the ring of Eisenstein integers is the cyclic group formed by the sixth roots of unity in the complex plane.
		y) and because of that it is not prime in the Eisenstein integers.
	en.wikipedia.org /wiki/Eisenstein_integer (356 words)

	Irreducibility Criteria
		Notice that Eisenstein's criterion essentially reduces the problem of factoring a polynomial of degree d to a problem of factoring d integers, the coefficients of the transformed polynomial, to see if they share a suitable common prime divisor.
		The only integer divisors of 2 are +-1 and +-2, so if the numbers k1 and k2 differ by more than 2, and if the absolute values of g(k1) and g(k2) are both 1, then g(k1) and g(k2) must have the same sign.
		Therefore, the maximum number of integers k differing by more than 2 for which g(k)=+-1 is only d_g, and the maximum number of such integers for which h(k)=+-1 is only d_h, for a combined total of (d_g + d_h) = d.
	www.mathpages.com /home/kmath406.htm (917 words)

	PlanetMath: Gaussian integer
		Using the ring of Gaussian integers, it is not hard to show, for example, that the Diophantine equation
		This is version 6 of Gaussian integer, born on 2001-10-15, modified 2003-10-08.
		For instance, they are a subring of the complex numbers, their graph on the complex plane forms a lattice of "squares", and they are a Euclidean domain.
	planetmath.org /encyclopedia/GaussianIntegers.html (152 words)

	Ferdinand Eisenstein - Wikipedia, the free encyclopedia
		Ferdinand Gotthold Max Eisenstein (April 16, 1823 - October 11, 1852) was a German mathematician.
		Like Galois and Abel, Eisenstein died before the age of 30, and like Abel, his death was due to tuberculosis.
		Gauss's choice of Eisenstein, who specialized in number theory and analysis, may seem puzzling to many, but it is justified by the fact that Eisenstein easily proved several results that were unattainable even for Gauss, like the theorem on biquadratic reciprocity.
	en.wikipedia.org /wiki/Ferdinand_Eisenstein (183 words)

	Arithmetic and lattices of imaginary quadratic rings (Site not responding. Last check: 2007-10-31)
		The maximal rings of quadratic integers of the imaginary quadratic fields are the subject of this paper.
		. This is unity with the Eisenstein integers.
		This diagram depicts the geometric construction of the conjugate of an Eisenstein integer laid over the (fundamental) lattice of the Gaussian integers on the complex plane. The other imaginary quadratic integers, congruent 1 (mod 4) are similar except the imaginary axis’s angle of inclination to the real axis is greater.
	paul-mccarthy.us /Misc/QuadArith.htm (760 words)

	Elliptic and Modular Functions
		Let z be a point in the upper half-plane and let L be a lattice in C. The Eisenstein series are defined as the coefficients of the Laurent Series expansion of the Weierstrass wp-function: wp(z, L) = ((1)/(z^2)) + sum_(2 <= k) G_k(L)(2k - 1)z^(2k - 2) where G_k(L) are the Eisenstein series.
		Given a positive even integer k = 2n and a lattice L = [a, b] in the complex plane, return the value of the Eisenstein series E_(2n)(z) relative to the lattice L. Eisenstein(k, F) : RngIntElt, QuadBinElt -> RngSerElt
		Given a pair L = [a,b] of complex numbers generating a lattice in C, return the normalized q-series expansion of the discriminant Delta(q) evaluated at tau where tau = a/b or tau = b / a, whichever is in the upper half complex plane.
	www.umich.edu /~gpcc/scs/magma/text570.htm (1235 words)

	Informat.io on Gaussian Integer
		A Gaussian integer is a complex number whose real and imaginary part are both integers.
		If the norm of a Gaussian integer z is a prime number, then z must be a Gaussian prime, since every non-trivial factorization of z would yield a non-trivial factorization of the norm.
		The ring of Gaussian integers is the integral closure of Z in the field of Gaussian rationals Q(i) consisting of the complex numbers whose real and imaginary part are both rational.
	www.informat.io /?title=gaussian-integer (395 words)

	Creation of Local Rings and Fields
		Given a prime integer p and non-negative single-precision integer k, construct the bounded free precision ring (field) of p-adic integers with maximum precision k.
		The polynomial f must be an Eisenstein polynomial, that is, the leading coefficient is a unit, the constant coefficient has valuation 1 and all other coefficients have valuation greater than or equal to 1.
		Given a non-negative single precision integer k, the map m must return the defining polynomial of the extension to precision k, as a polynomial over R. The map m's behaviour for other input values is undefined.
	www.math.lsu.edu /magma/text747.htm (1665 words)

	Algebraic integer - ExampleProblems.com
		In mathematics, an algebraic integer is an algebraic number that is a root of a monic polynomial (i.e.
		So all radical integers are algebraic integers but not all algebraic integers are radical integers.
		In other words, the algebraic integers form a ring that is closed under the operation of extraction of roots.
	www.exampleproblems.com /wiki/index.php/Algebraic_integer (183 words)

	id:A112770 - OEIS Search Results
		Eisenstein integers are of the form a + bomega, where a and b are ordinary integers, and omega = (-1 + isqrt(3))/2 is a cube root of 1, the other cube roots of 1 being 1 and omega^2 = (-1 - i*sqrt(3))/2.
		The analogue of Fermat's 4n+1 Theorem for Eisenstein integers is that a prime p can be written in the form a^2 - ab + b^2 = (a + bomega)(a + bomega^2) iff 3 does not divide (p+1).
		Eisenstein integers are complex numbers that are also members of the imaginary quadratic field Q(sqrt -3) = Z[omega].
	www.research.att.com /~njas/sequences/A112770 (284 words)

	RCF - Book Reviews
		Sam Eisenstein marvels at the ancient itch of lust, understands that its enticing energy effortlessly structures and then detonates private universes.
		Eisenstein marvels at the sheer breadth of its impact, how it slams gently into consciousness, how its subtle terrorism un-works, un-clarifies.
		With the heart of a Beat poet and the soul of a medieval divine, Eisenstein knows that lust translated into dreary backyard realism thins its vicious splendor into pedestrian urges and inevitably reduces fiction to greeting-card platitudes or talk-show psycho-blather.
	www.centerforbookculture.org /review/bookreviews/04_2/eisenstein.html (522 words)

	Reciprocity Laws. Rule of Quadratic Reciprocity - Numericana
		Eisenstein's Lemma: A variation of Gauss's lemma allowing a simpler proof.
		For a given odd prime p, let's denote < a > the integer congruent to a which is between -p/2 and +p/2.
		Since the result is trivial when p divides a (the sign of 0 being 0) we may assume that a is invertible modulo p.
	home.att.net /~numericana/answer/reciprocity.htm (1523 words)

	Creation of Local Rings and Fields
		The parameters describing a local ring or field are: the characteristic p of its residue class field, the inertia degree f or inertial polynomial f, the Eisenstein polynomial g of degree e describing the totally ramified extension of the inertia ring and the precision n.
		The ring R of integers in the local field L. This is the set of elements of non-negative valuation.
		So an element of a p-adic field is printed as an integer co--prime to the prime of the field multiplied by the prime to the power of the valuation of the element.
	www.umich.edu /~gpcc/scs/magma/text708.htm (2218 words)

	[No title]
		Z[omega] is the medium-well-known ring of Eisenstein integers, which has unique factorization (not to mention the Euclidean algorithm) just as the Gaussian integers do.
		It is fortuitous that the Eisenstein integers are a PID, because it means that irreducible numbers (n = ab ==> na or nb) are prime (nab ==> na or nb).
		Finally, 3 is not an Eisenstein prime, but since has only one Eisenstein prime factor that appears twice, rather than two distinct ones, its presence complicates the set of solutions but it does not produce solutions ex nihilo.
	www.math.niu.edu /~rusin/known-math/93_back/calc.int (889 words)

	Vita
		The general philosophy is that if a `nice' function from number theory is forced to be zero at its center, its derivative at the center should be very interesting and might be linked to arithmetic of a geometric subject.
		There is a systematic way to produce Eisenstein series on $Sp(n)$ which vanishes at the center from `incompatible' local quadratic space system of dimension $n+1$.
		Kudla has a general scheme to compute the central derivative of such Eisenstein series, and conjecture that its restriction on subgroups of $Sp(n)$ (by doubling method for example) are closely related to the global intersection numbers of cycles on Shimura varieties arising naturally.
	www.math.wisc.edu /~thyang/publ.html (1805 words)

	ARCC Workshop: Eisenstein Series and Applications (Site not responding. Last check: 2007-10-31)
		Eisenstein series are an essential ingredient in the spectral theory of automorphic forms and an important tool in the theory of L-functions.
		But the Eisenstein series themselves are often treated as auxillary objects.
		A central goal of the workshop will be to try to understand the common structural properties of the Eisenstein series occuring in these and related applications.
	aimath.org /ARCC/workshops/eisenstein.html (412 words)

	Number Theory
		For the case k(rho), written here as k(r), (also known as the "Eisenstein integers"), the notion of conjugate is a bit different.
		Both of these equations (1) and (2) being satisfied by Gaussian and Eisenstein integers, resp., have a and b ordinary or "rational" integers so the Gaussian and Eisenstein integers are quadratic integers as defined in H & W, that is, they satisfy quadratic equations with integer coefficients.
		Also for 1.2, find an integer value of k (at least 4) so that, converting the expansion of pi with respect to k, we find a perfect square in one of the truncations.
	www.georgetown.edu /faculty/kainen/numbertheory.html (3931 words)

	Minkowski biography
		Eisenstein had given a formula for the number of such representations in 1847, but he had not given a proof of the result.
		Eisenstein had been studying quadratic forms in n variables with integer coefficients at the time he published his unproved formula in 1847 but as he was already ill by this time details were never published.
		Minkowski, although only eighteen years old at the time, reconstructed Eisenstein's theory of quadratic forms and produced a beautiful solution to the Grand Prix problem.
	www-history.mcs.st-and.ac.uk /history/Biographies/Minkowski.html (1536 words)

	PrimeFan's Listing of Esoteric Integer Sequences
		Smallest composite number such that the next n - 1 integers have the property that the list of smallest prime factors not in common with a smaller number of the set is the set of the first n prime numbers.
		Smallest composite number such that the previous n - 1 integers have the property that the list of smallest prime factors not in common with a smaller number of the set is the set of the first n prime numbers.
		Sequences in Sloane's Online Encyclopedia of Integer Sequences in which the A number is not a member of the sequence (leading zeroes ignored).
	www.geocities.com /primefan/EsotericIntegerSequences.html (4668 words)

	Welcome to Red Hen Press
		Sam Eisenstein is the author of numerous novels and plays including, The Inner Garden (Sun & Moon Press, 1987), Prince of Admission (Sun & Moon Press) and Rectification of Eros: Havana (Green Integer, 1999).
		Eisenstein graduated from UCLA in 1995 with a PHD in comparative literature.
		In 1973, Eisenstein earned his MA in Psychology from Goodaed College and became a licensed marriage and family counselor in 1974.
	www.redhen.org /authorDetail.asp?authorID=70 (309 words)

	Electrons Favor One Direction over Others
		When Jim Eisenstein, now of the California Institute of Technology, and his colleagues observed strange direction-dependent resistance in quantum Hall data more than a decade ago, "we thought it was some kind of garbage," Eisenstein recalls.
		The results remain unexplained, although theorists are favoring a scenario based on a collective state of strongly interacting electrons--akin to that in the fractional quantum Hall effect--but one where a single direction in space is preferred.
		How that direction gets chosen is "the big bugaboo, the thing that nobody has a clue about," says Eisenstein, although he assumes it is some very weak effect, just as a knitting needle balanced on its tip will fall in the direction favored by the slightest vibration or material imperfection.
	focus.aps.org /story/v3/st2 (713 words)

	On Quaternions and Octonions, by John Conway and Derek Smith
		The Gaussian and Eisenstein integers are the most symmetrical lattices in the plane, since they have 4-fold and 6-fold rotational symmetry, respectively.
		The Lipschitz integers are a subring of the quaternions, and this has a nice application to ordinary number theory.
		The double Hurwitzian integers are closed under multiplication, and it is easy to see that as a lattice, they are the product of two copies of the Hurwitz integers -- hence their name.
	math.ucr.edu /home/baez/octonions/conway_smith (4511 words)

	LASEC
		The ring of Eisenstein integers is the set of all complex numbers which can be written a+jb where a and b are regular integers.
		The ring of Gauss integers is the set of all complex numbers which can be written a+ib where a and b are regular integers.
		The special d=3 or d=4 cases with Eisenstein or Gauss integers have the interesting property to lead to a 2-level secret (namely the secret sigma and the secret factorization).
	lasecwww.epfl.ch /memo/memo_mova.shtml (4887 words)

	Question on mathematics of ECC scheme
		(a,b) represent the quadratic integer a + b*w.
		integers Z[j] mod p, i.e., a + bj mod p, where j is defined by
		What I mean is 'Eisenstein integers of which the "coefficients"
	www.groupsrv.com /science/about8607.html (1260 words)

	Eisenstein versus Gauss (Site not responding. Last check: 2007-10-31)
		Gauss himself considered his third proof to be the most direct and natural of his demonstrations.
		While Eisenstein essentially follows the same outline as Gauss, each feature of his approach displays great clarity and insight, and offers an elegant view while shortening the path taken by Gauss.
		While Eisenstein's algebraic exponent is easily transformed into the exponent in (4) via (2), Gauss must establish a number of technical properties of the greatest integer function and apply them to relate
	www.math.nmsu.edu /~history/eisenstein/node3.html (389 words)

	[No title]
		This procedure may well cause violations of algebraic # rules one would expect, such as that long division of a into b should # give the same result as long division of ca into cb, whenever c is a # unit, and should give the same quotient for any c.
		At each stage you multiply the old matrix by the # simple matrix [[q,1],[1,0]] where q is the quotient in the simple gcd # algorithm.
		Of course, with our representation of Eisenstein integers # as pairs, we can't just call up the library routines for matrix # multiplication in the E-context.
	www.math.tamu.edu /~Doug.Hensley/eisenfix.txt (307 words)

	Mathematics Itself: On the Origin, Nature, Fabrication of Logic and Mathematics (Site not responding. Last check: 2007-10-31)
		One would assume that these kinds of numbers are “kinds of Integers.” However, a Gaussian integer is not a “Real number” as all Integers are, but is a “Complex number.” A Gaussian integer is complex number in the form a + bi where a and b are integers and i= sqrt(-1).
		An Eisenstein integer is even more confusing because it is defined a + bw, where a and b are integers and w=(-1- sqrt(-7))/2.
		Even integers are limited to an approximation within the computer (that is, each integer has an "approximating" finite limit of bits for representation, which works as long as the numbers don't exceed the limit).
	users.viawest.net /~keirsey/mathitself.html (10337 words)

	Eisenstein Series
		and r is an integer with r > 2.
		Furthermore, each Eisenstein series is expressible as a polynomial of the elliptic invariants
		Sloane, N. Sequences A001067, A004009/M5416, A004011/M5140, A006863/M5150, A008410, A013973, and A013974 in "The On-Line Encyclopedia of Integer Sequences."
	users.skynet.be /fa956617/math/topics/EisensteinSeries.html (245 words)

	Eisenstein's Proof (Site not responding. Last check: 2007-10-31)
		for any integer b not divisible by p, holds because the nonzero residue classes form a (cyclic) group of order p-1 under multiplication.
		Here is Eisenstein's proof, closely following both his own language and notation (which he conveniently and successfully abuses).
		Eisenstein now uses a geometric representation of the exponent in this last equation to transform it twice while retaining its parity: This exponent is precisely the number of integer lattice points with even abscissas lying in the interior of triangle ABD in the Figure (note that no lattice points lie on the line AB).
	www.math.nmsu.edu /~history/eisenstein/node2.html (475 words)

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