Gaussian rationals

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Topic: Gaussian rationals

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	[ref] 56 Vector Spaces
		A vector space is Gaussian if it allows Gaussian elimination; this is used for row vector spaces and matrix vector spaces.
		This operation is defined only for semi-echelonized bases (or mutable bases) of Gaussian row and matrix vector spaces.
		gap> V:= FullRowSpace( Rationals, 2);; gap> W:= VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ]);; gap> H:= Hom( Rationals, V, W); Hom( Rationals, ( Rationals^2), VectorSpace( Rationals, [ [ 1, 0, 1 ], [ 1, 2, 3 ] ])) gap> Dimension( H); 4
	adela.karlin.mff.cuni.cz /kam/netkarl/gap/ref/CHAP056.htm

	GAP Manual: 14 Gaussians
		The field of Gaussian rationals is just a special case of cyclotomic fields, so everything that applies to those fields also applies to it (see chapters Cyclotomics and Subfields of Cyclotomic Fields).
		This field is called the Gaussian rationals and its ring of integers is called the Gaussian integers, because C.F. Gauss was the first to study them.
		Of course either operand may also be an ordinary rational (see Rationals), because the rationals are embedded into the Gaussian rationals.
	www.institut.math.jussieu.fr /~jmichel/htm/CHAP014.htm

	Ch11-ComputingInFields.html
		Case 1: The Gaussian rationals, Q[I] We start with the Gaussian rationals, a field in which we know how to compute.
		This rule for inverses is specific to the Gaussian rationals.
		Our general theorem has L = K[x]/(f(x)) being a field, whenever K is a field and f(x) is an irreducible polynomial over K. So far we have been looking at the case when K is Q, the field of rationals.
	www.adeptscience.co.uk /products/mathsim/maple/powertools/abstractalgebra/html/Ch11-ComputingInFields.html

	Gaussian rational - Wikipedia, the free encyclopedia
		The field of Gaussian rationals is neither ordered nor
		field of Gaussian rationals is the field Q ( i) formed by adjoining the imaginary number i to the field of
		The Gaussian integers form the ring of integers of Q ( i).
	en.wikipedia.org /wiki/Gaussian_rational

	Gaussian rational - Wikipedia, the free encyclopedia
		field of Gaussian rationals is the field Q ( i) formed by adjoining the imaginary number i to the field of
		The field of Gaussian rationals is neither ordered nor
		The Gaussian integers form the ring of integers of Q ( i).
	en2.wikipedia.org /wiki/Gaussian_rational

	gaussian.g
		Gaussian rationals are of the form ' + \E(4)', ## where and are rationals*.
		is a Gaussian integer and## 'false' otherwise.
		is a Gaussian ## integer and 'false' otherwise.
	www.ece.utexas.edu /projects/ljohn/ravi/other_inputs/gap/gaussian.g

	An-Az
		It provides a C++ wrapper around it and extends it in a consistent mathematical way to deal with results of operations that cannot be represented as rationals or are mathematically undefined.
		rationals, fractions, vectors, matrices and linear algebra, polynomials and algebraic numbers.
		The rational class library of Arithmos is built on top of the GMP rational C library.
	stommel.tamu.edu /~baum/linuxlist/linuxlist/node8.html

	Algebraic number
		In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.
		The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a ring.
		This can be rephrased by saying that the field of algebraic numbers is algebraically closed.
	www.sciencedaily.com /encyclopedia/algebraic_number

	Jim Loy's Mathematics Page
		From natural numbers, it can be generalized to rationals, as fractions with interesting numerators and denominators are obviously interesting.
		And what could be more interesting than an irrational that cannot be formed from any finite combination of rationals?
		And the proof that all numbers are interesting should not be boring.
	www.jimloy.com /math/math.htm

	Gaussian integer - Wikipedia, the free encyclopedia
		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.
		Gaussian Integers, Fermat's Last Theorem Blog traces the history of Fermat's Last Theorem from Diophantus of Alexandria to Andrew Wiles.
		Those rational primes which are congruent to 3 (mod 4) are Gaussian primes; those which are congruent to 1 (mod 4) are not.
	www.wikipedia.org /wiki/Gaussian_integer (331 words)

	Gaussian integer - Wikipedia, the free encyclopedia
		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.
		Those rational primes which are congruent to 3 (mod 4) are Gaussian primes; those which are congruent to 1 (mod 4) are not.
		Some prime numbers (which, by contrast, are sometimes referred to as "rational primes") are not Gaussian primes; for example 2 = (1 + i)(1 − i) and 5 = (2 + i)(2 − i).
	en.wikipedia.org /wiki/Gaussian_integer (331 words)

	gaussian prime
		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.
		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.
		A Gaussian integer is a complex number whose real and imaginary part are both integers.
	www.yourencyclopedia.net /Gaussian_prime (331 words)

	Gaussian integer - Wikipedia, the free encyclopedia
		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.
		A Gaussian integer is a complex number whose real and imaginary part are both integers.
		Gaussian Integers, Fermat's Last Theorem Blog traces the history of Fermat's Last Theorem from Diophantus of Alexandria to Andrew Wiles.
	en.wikipedia.org /wiki/Gaussian_integer (331 words)

	GAP Manual: 1.28 About Defining New Domains
		For further details see chapter Gaussians for a description of the Gaussian integers and rationals and chapter Rings for a list of all functions applicable to rings.
		A rational integer is a prime in the ring of Gaussian integers if and only if it is congruent to 3 modulo 4 (the other rational integer primes split into two irreducibles), and a Gaussian integer that is not a rational integer is a prime if its norm is a rational integer prime.
		For Gaussian integers we return that associate that lies in the first quadrant of the complex plane.
	www-groups.dcs.st-and.ac.uk /gap/Gap3/Manual3/C001S028.htm (331 words)

	Nineteenth Century Geometry
		In his study of curved surfaces, Gauss introduced a real-valued function, the Gaussian curvature, which measures a surface's local deviation from flatness in terms of the surface's intrinsic geometry.
		, was supplemented with zero, the negative integers, the non-integral rationals, the irrationals, and the so-called imaginary numbers.
		However, it had been practised in arithmetic for centuries, as the initial stock of natural numbers 1, 2, 3,...
	adt.library.usyd.edu.au /stanford/entries/geometry-19th (331 words)

	GAP Manual: 1.28 About Defining New Domains
		For further details see chapter Gaussians for a description of the Gaussian integers and rationals and chapter Rings for a list of all functions applicable to rings.
		A rational integer is a prime in the ring of Gaussian integers if and only if it is congruent to 3 modulo 4 (the other rational integer primes split into two irreducibles), and a Gaussian integer that is not a rational integer is a prime if its norm is a rational integer prime.
		For Gaussian integers we return that associate that lies in the first quadrant of the complex plane.
	www-groups.dcs.st-and.ac.uk /gap/Gap3/Manual3/C001S028.htm (1933 words)

	GAP Manual: 4 Domains
		gap> GaussianIntegers < Rationals; Error, sorry, cannot compare with the infinite domain gap> Group( (1,2), (1,2,3,4,5,6)) < D12; true # since '(5,6)', the second element of the left operand, # is less than '(2,6)(3,5)', the second element of 'D12'.
		For example suppose you want to factor 10 in the ring of Gaussian integers.
		For example the ring of Gaussian integers is predefined as
	www.institut.math.jussieu.fr /~jmichel/htm/CHAP004.htm (1933 words)

	jacobi
		pari is pretty good, but only within its limited scope -- it is _not_ a symbolic algebra program, although it is very good with integers, rationals, polynomials, etc. I have found at least one bug -- gcd() doesn't work correctly over the Gaussian integers (and probably not over other algebraic integral domains, either, I suspect).
		Jacobi's symbol is the generalization of (a/b) to all odd b, but it loses some of the usefulness for non-prime b, since it no longer correctly identifies non-residues.
		Maple's problem is in not clearly distinguishing between their function L(a,b) and Legendre's symbol (a/b), which is defined _only_ for prime b.
	www.math.niu.edu /~rusin/known-math/97/jacobi (1550 words)

	ram2.txt
		This paper shows that such a splitting at the E1 level does not occur for * p = 2 when R is the ring of integers in a quadratic extension of the rationals* which ramif* *ies at 2.
		The Topological Hochschild Homology of the Gaussian Integers Ayelet Lindenstrauss Department of Mathematics The Technion Haifa 32000, Israel x0.
		Topological Hochschild homology can also be defined for more general spectr* *a than those arising from discrete rings ([15], [9]).
	hopf.math.purdue.edu /Lindenstrauss/ram2.txt (1550 words)

	Free Numerical, Mathematical and Statistical Libraries and Source Code (thefreecountry.com)
		Among the numerous functions available are: the elementary functions (log, sin, cos, exp, etc), gamma, psi, dilogarithm, Airy, Bessel, hypergeometric, Struve, complete and incomplete elliptic functions, Planck radiation, Fresnel integrals, probability integrals and their inverses (Gaussian, Poisson, F, Chi-square, binomial, Kolmogorv-Smirnov arithmetic on polynomials, rationals, etc), and so on.
		Cephes is a C library that provides numerous mathematical functions including support for the complex variable types that will be in the new ANSI C standard, C99.
		GSL is a numerical library that may be used in C and C++ programs.
	www.thefreecountry.com /sourcecode/mathematics.shtml (1550 words)

	Exact Solution of Linear Equation Systems over Rational Numbers by Parallel p-Adic Arithmetic - Limongelli, Pirastu (ResearchIndex)
		The rationals are represented by truncated p-adic expansion.
		We describe a parallel implementation of an algorithm for solving systems of linear equations over the field of rational numbers based on Gaussian elimination.
		The parallelization is based on a multiple homomorphic image technique and the result is recovered by a parallel version...
	citeseer.ist.psu.edu /limongelli94exact.html (1550 words)

	jacobi
		pari is pretty good, but only within its limited scope -- it is _not_ a symbolic algebra program, although it is very good with integers, rationals, polynomials, etc. I have found at least one bug -- gcd() doesn't work correctly over the Gaussian integers (and probably not over other algebraic integral domains, either, I suspect).
		Jacobi's symbol is the generalization of (a/b) to all odd b, but it loses some of the usefulness for non-prime b, since it no longer correctly identifies non-residues.
		I don't know the implementation of maple's Legendre function, but if it is the same Legendre/Jacobi function that I know, it does _not_ require factorization, and should be of the same or slightly easier complexity than gcd(p,q).
	www.math.niu.edu /~rusin/known-math/97/jacobi (1550 words)

	Root of unity
		Conversely every abelian extension of the rationals is a subfield of a cyclotomic field - theorem of Kronecker usually called the Kronecker-Weber theorem on the grounds that Weber supplied proof.
		In cases Galois theory can be written out explicitly in terms of Gaussian periods : this theory from the Disquisitiones Arithmeticae of Gauss was published many years before Galois.
		This work is essentially an ecumenical (emphasis on ecumenical) dialogue between theologians from various nations and christian traditions concerning the topic of baptism.
	www.freeglossary.com /Root_of_unity (1550 words)

	Root of unity
		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.
		In these cases Galois theory can be written out quite explicitly in terms of Gaussian periods : this theory from the Disquisitiones Arithmeticae of Gauss was published many years before Galois.
		One may summon his philosophy when they are beaten in battle, not till then.
	www.brainyencyclopedia.com /encyclopedia/r/ro/root_of_unity.html (1550 words)

	jacobi
		pari is pretty good, but only within its limited scope -- it is _not_ a symbolic algebra program, although it is very good with integers, rationals, polynomials, etc. I have found at least one bug -- gcd() doesn't work correctly over the Gaussian integers (and probably not over other algebraic integral domains, either, I suspect).
		Jacobi's symbol is the generalization of (a/b) to all odd b, but it loses some of the usefulness for non-prime b, since it no longer correctly identifies non-residues.
		Maple's problem is in not clearly distinguishing between their function L(a,b) and Legendre's symbol (a/b), which is defined _only_ for prime b.
	www.math.niu.edu /~rusin/known-math/97/jacobi (1550 words)

	Talking Rationally About Surrogate Factoring
		Now, any non-zero rational number r is a factor of an integer N in the ring of rationals, so he wants to make a differentiation between "trivial" rational factors (which act like the integer 1, M), and "non-trivial" rational factors (which act like the proper divisors of M).
		This doesn't seem likely; the only example I can think of right off is of Gaussian elimination, where results of previous row operations determine further operations.
		The result of the n^3 operations on a 0-1 matrix is that one entry in the matrix has a denominator whose size is exponential in n, the size of the matrix.
	www.talkabouteducation.com /group/alt.math/messages/29854.html (1233 words)

	ram26.txt
		Introduction This paper gives an explicit calculation of the 2-torsion in the topologica* l Hochschild homology THH of rings of integers in quadratic extensions of the rationals* wh* *ich are ramified at the prime 2 (Theorem (1.14)).
		The Topological Hochschild Homology of the Gaussian Integers Ayelet Lindenstrauss Department of Mathematics The Technion Haifa 32000, Israel ayeletl@tx.technion.ac.il x0.
		We represent a2 by the homology class a2 of the 4-dimensional chain (1 ^ i_* *^ i_) x 2 + (1 ^ (M1 ^ _)) x 1 on THH (2)(R), for M1 of (3.3.1).
	hopf.math.purdue.edu /Lindenstrauss/ram26.txt (1233 words)

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Topic: Gaussian rationals

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