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Topic: Ring of integers

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integer factorization

Integer computer science

positive integers

algebraic integers

Integer partition

Gaussian integer

Integer sequences

p adic integers

On Line Encyclopedia of Integer Sequences

ring of integers

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	Ring of integers - Wikipedia, the free encyclopedia
		the ring of all integers (whole numbers: positive, negative or zero), usually denoted by Z.
		since Z is the ring of integers of the field Q of rational numbers.
		And indeed, in algebraic number theory the elements of Z are often called the "rational integers" because of this.
	en.wikipedia.org /wiki/Ring_of_integers (203 words)

	Ring (mathematics) - Wikipedia, the free encyclopedia
		A ring is a generalization of the integers, which itself is an example of a ring.
		Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings.
		The motivating example is the ring of integers with the two operations of addition and multiplication.
	en.wikipedia.org /wiki/Ring_(mathematics) (1408 words)

	Ring Theory
		The decomposition of an integer into the product of powers of primes has an analogue in rings where prime integers are replaced by prime ideals but, rather surprisingly, powers of prime integers are not replaced by powers of prime ideals but rather by "primary ideals".
		However, axioms for rings are not given by Weber and the axiomatic treatment of commutative rings was not developed until the 1920's in the work of Emmy Noether and Krull.
		In contrast to commutative ring theory, which as we have seen grew from number theory, non-commutative ring theory developed from an idea which, at the time of its discovery, was heralded as a great advance in applied mathematics.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Ring_theory.html (1890 words)

	[No title]
		The ideal in the free ring on the letters a, b generated by ab-ba: kernel This symbol represents a unary function.
		The kernel of a ring homomorphism is an ideal.
		The first argument is a ring R and the second argument is an element of R. When evaluated on R and such a second argument, the function represents the ideal in R generated by the second argument.
	www.win.tue.nl /~amc/oz/om/cds/ring3.html (391 words)

	On p-adic Arithmetic: p-adic Integers and p-adic Numbers - Numericana
		Decadic Integers: The strange realm of 10-adic integers (composite radix).
		The field of p-adic numbers is the quotient field of the ring of p-adic integers.
		The field of p-adic numbers is to the ring of p-adic integers what the field of rationals is to the ring of ordinary integers: More precisely, the p-adic numbers form the quotient field of the ring of p-adic integers.
	home.att.net /~numericana/answer/p-adic.htm (3746 words)

	Springer Online Reference Works
		is an invertible element of the ring of
		A necessary condition for the solvability of this equation in integers or in rational numbers is its solvability in the rings or, correspondingly, in the fields of
		This theorem, which is known as the Hensel lemma, is a special case of a more general fact in the theory of schemes.
	eom.springer.de /P/p071020.htm (465 words)

	Rings (Site not responding. Last check: 2007-10-01)
		Rings are an extension of groups, and you should be familiar with normal subgroups and group homomorphisms in order to proceed with rings.
		in fact a ring is a group and a monoid acting on the same set cooperatively.
		The integers form a ring and the rationals form a division ring.
	www.mathreference.com /ring,intro.html (280 words)

	Creation Functions (Site not responding. Last check: 2007-10-01)
		Create the ring of integers Z. Analogous to the creation of the ring of integers of any number field, there is a version of IntegerRing that creates Z as the ring of integers of Q. Creation of Elements
		Since the ring of integers is around when Magma is started up, literally typed integers will be regarded as elements of the ring of integers.
		Given a positive integer b, and a sequence s = [ a_1,..., a_n ] of positive integers such that 0 <= a_i < b, construct the integer a = a_1b^0 + a_2b^1 +...
	www.math.ufl.edu /help/magma/text310.html (359 words)

	Rings
		A Division Algebra is a nontrivial ring (not necessarily commutative) in which all nonzero elements are invertible.
		The kernel of a homomorphism of rings f: A --> B is the ideal in A consisting of those elements a in A such that f(a) = 0.
		An ideal J in a ring A is prime if the elements of A not in J are closed under multiplication.
	mcraefamily.com /MathHelp/BasicAARings.htm (1006 words)

	[ref] 14 Integers
		integers are entered as a sequence of decimal digits optionally preceded by a '+' sign for positive integers or a '-' sign for negative integers.
		Many more functions that are mainly related to the prime residue group of integers modulo an integer are described in chapter Number Theory, and functions dealing with combinatorics can be found in chapter Combinatorics.
		The residue class rings are rings, thus all operations for rings (see Chapter Rings) apply.
	www.math.niu.edu /help/math/gap4/ref/CHAP014.htm (1290 words)

	Emmy Noether (Site not responding. Last check: 2007-10-01)
		In particular, she is viewed as one of the pioneers of what today are called "ring theory" and "ideal theory." The concept of "Noetherian Rings" plays a central role in the study of abstract algebra.
		Can you find more ideals of the ring Z? In the next lessons we will determine what all the ideals of the ring of integers look like, describe what "Noetherian" means and show that the ring Z is a Noetherian ring.
		The significance of Emmy Noether's contributions lies in the fact that her results are not only true for the integers, but also hold for a far more general class of structures that have the property of being a ring.
	www.math.wsu.edu /math/faculty/barbut/Ring_module.htm (510 words)

	Polynomial Extensions of Finite Rings (Site not responding. Last check: 2007-10-01)
		As with all rings, the additive group is abelian.
		In this case, the additive group is isomorphic to the ring R
		The quotient ring is a field if the modulus polynomial is irreducible.
	www.hostsrv.com /webmaa/app1/MSP/webm1010/PolyExtensionRing.msp (83 words)

	Unique and nonunique factorization
		We define a ring of integers to be the set of algebraic integers in a number field.
		We say a ring of integers has unique factorization if whenever an element of a ring of integers is expressed as a product of irreducible elements, that expression is unique up to changing the order and multiplying by units.
		Theorem 4 (Minkowski) The number of equivalence classes of ideals in a ring of integers is finite.
	mathcircle.berkeley.edu /BMC3/bamc/node2.html (647 words)

	Creation Functions
		Since the ring of integers is present when Magma is started up, integers typed into Magma without any explicit context will be regarded as elements of the ring of integers.
		Given a non-negative integer n and a positive integer b >= 2, return the unique base b representation of n in the form of a sequence Q. That is, if n = a_0b^0 + a_1b^1 +...
		Given a positive integer b >= 2 and a sequence Q = [ a_0,..., a_(k - 1) ] of non-negative integers such that 0 <= a_i < b, return the integer n = a_0b^0 + a_1b^1 +...
	www.umich.edu /~gpcc/scs/magma/text530.htm (428 words)

	Algebraic Integers
		Algebraic integers are roots of a polynomial with integral coefficients and leading coefficient 1.
		For example, the algebraic integers of degree 1 are the ordinary integers (elements of Z).
		Exercise An algebraic integer a of degree d is root of a polynomial with integral coefficients of degree d.
	www.win.tue.nl /~aeb/an/an.html (584 words)

	Rings, Unit Rings, Ideals, Integral Domains and Fields - Numericana (Site not responding. Last check: 2007-10-01)
		Rings are sets endowed with addition and multiplication,
		An ideal is a subset of a ring closed under addition and multiplication.
		Residue Ring (modulo a given ideal I of a ring A) A/I is.
	home.att.net /~numericana/answer/ring.htm (130 words)

	Rings of Algebraic Integers (Site not responding. Last check: 2007-10-01)
		of all algebraic integers is a ring, i.e., the sum and product of two algebraic integers is again an algebraic integer.
		be the ring of integers of a number field.
		In the next two sections we will develop some basic properties of norms and traces, and deduce further properties of rings of integers.
	modular.fas.harvard.edu /129/ant/html/node14.html (246 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-01)
		Now 3 is not expressible in the form a^2 + b^2, so it is not composite in the ring of Gaussian integers.
		It is also possible to prove that every prime number of the form 4k + 1 is expressible as the sum of two squares, and no prime number of the form 4k + 3 is expressible in that way.
		That tells us that the prime integers which are composite in the Gaussian integers are those not leaving remainder 3 when divided by 4.
	mathforum.org /dr.math/problems/scomazzon8.11.98.html (469 words)

	Math Forum Discussions (Site not responding. Last check: 2007-10-01)
		non-contradiction with coprimeness results in the ring of integers.
		integers, and include the integers, then you have what you need.
		So you have the ring of integers and the ring of objects, which
	mathforum.org /kb/thread.jspa?threadID=1094146&messageID=3551832 (172 words)

	GAP Manual: 10. Integers
		The size of integers in GAP is only limited by the amount of available memory, so you can compute with integers having thousands of digits.
		Since the integers form a Euclidean ring all the ring functions are applicable to integers (see chapter Rings, Ring Functions for Integers, Primes, IsPrimeInt, IsPrimePowerInt, NextPrimeInt, PrevPrimeInt, FactorsInt, DivisorsInt, Sigma, Tau, and MoebiusMu).
		Since the integers are naturally embedded in the field of rationals all the field functions are applicable to integers (see chapter Fields and Field Functions for Rationals).
	www.math.uiuc.edu /Software/GAP-Manual/Integers.html (262 words)

	Gaussian integers (Site not responding. Last check: 2007-10-01)
		Geometrically, addition of Gaussian integers may be visualized using the parallelogram law: if you represent the integer
		There is an analog of the fundamental theorem of arithmetic for the ring of Gaussian integers.
		St] (and independently by A. Baker) that the only cases for which ring of integers of
	web.usna.navy.mil /~wdj/book/node56.html (215 words)

	The Ring of Integers, Euclidean Rings and Modulo Integers
		In this article we introduce the ring of Integers, Euclidean rings and Integers modulo $p$.
		In particular we prove that the Ring of Integers is an Euclidean ring and that the Integers modulo $p$ constitutes a field if and only if $p$ is a prime.
		The divisibility of integers and integer relatively primes.
	mizar.uwb.edu.pl /JFM/Vol11/int_3.html (174 words)

	The ring of integers mod 8 (Site not responding. Last check: 2007-10-01)
		The ring of integers consists of the set {0, 1, 2, 3, 4, 5, 6, 7} where these eight numbers can be thought of being arranged in a circle as in the face of a clock:
		To add or multiply the numbers, simply follow the arrows and see where you land.
		Notice that although arithmetic in this ring is similar to that in the ring Z of integers, there are also some differences.
	www.math.wsu.edu /math/faculty/barbut/mod8.htm (169 words)

	Citebase - Additive number theory and the ring of quantum integers (Site not responding. Last check: 2007-10-01)
		Additive number theory and the ring of quantum integers
		These constructions lead to the construction of the ring of quantum integers and the field of quantum rational numbers.
		It is also shown that addition and multiplication of quantum integers are equivalent to elementary decompositions of intervals of integers in additive number theory.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0204006 (144 words)

	Rearrangements of functions on the ring of integers of a $p$-series field., Benjamin B. Wells
		Rearrangements of functions on the ring of integers of a $p$-series field., Benjamin B. Wells
		Rearrangements of functions on the ring of integers of a $p$-series field.
		[2] R. Hunt and M. Taibleson, Almost everywhere convergence of Fourier series on the ring of integers of a local field, SIAM J. Math.
	projecteuclid.org /getRecord?id=euclid.pjm/1102866969 (116 words)

	The Adele Ring (Site not responding. Last check: 2007-10-01)
		is the topological ring whose underlying topological space is the restricted topological product of the
		2 (Principal Adeles) The image of (20.3.2) is the ring of.
		The map is also a ring homomorphism, so the two sides are algebraically isomorphic, as claimed.
	modular.fas.harvard.edu /129/ant/html/node83.html (535 words)

	Efficient Algorithms for gcd and Cubic Residuosity in the Ring of Eisenstein Integers (Site not responding. Last check: 2007-10-01)
		We present simple and efficient algorithms for computing gcd and cubic residuosity in the ring of Eisenstein integers,
		The algorithms are similar and may be seen as generalisations of the binary integer gcd and derived Jacobi symbol algorithms.
		The technique underlying our algorithms can be used to obtain equally fast algorithms for gcd and quartic residuosity in the ring of Gaussian integers,
	www.brics.dk /BRICS/RS/03/8 (127 words)

	Topologies on the ring of integers of a global field., Jo-Ann Cohen
		Topologies on the ring of integers of a global field., Jo-Ann Cohen
		Topologies on the ring of integers of a global field.
		[14] W. Wieslaw, Locally bounded topologies on some Dedekind rings, Archiv der Math., 33 (1979), 41-44.
	projecteuclid.org /getRecord?id=euclid.pjm/1102736259 (152 words)

	Math 772
		In this assignment, we will do an intensive analysis of a particular ring of integers.
		Determine the number of real and complex embeddings of
		and its ring of integers is a UFD, then its set of norms factor uniquely (extra extra credit for the converse to this).
	www.ndsu.nodak.edu /ndsu/coykenda/M772.EX2.S2002.htm (127 words)

	Primes in ring of Gauss integers - help!! (Site not responding. Last check: 2007-10-01)
		Primes in ring of Gauss integers - help!!
		Apparently the first part of this problem applies, but i'll have to think about this more.
		Ah, so that's why you mentioned R. (Incidentally, I think you meant algebraic integers, not Gaussian integers)
	www.physicsforums.com /showthread.php?t=59946 (578 words)

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