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Topic: Unique factorization domain

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	PlanetMath: UFD
		On a UFD, the concept of prime element and irreducible element coincide.
		See Also: integral domain, irreducible, Euclidean domain, Euclidean valuation, proof that a Euclidean domain is a PID, motivation for Euclidean domains,
		This is version 12 of UFD, born on 2001-11-04, modified 2006-12-22.
	planetmath.org /encyclopedia/UniqueFactorizationDomain.html (138 words)

	Unique factorization domain - Wikipedia
		In mathematics, a unique factorization domain (UFD) is, roughly speaking, a ring in which every element can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers.
		The uniqueness part is sometimes hard to verify, which is why the following equivalent definition is useful: a unique factorization domain is an integral domain R in which every non-zero non-unit can be written as a product of prime elements of R.
		All principal ideal domains are UFD's; this includes the integers, all fields, all polynomial rings K[X] where K is a field, and the Gaussian integers Z[i].
	wikipedia.findthelinks.com /un/Unique_factorization_domain.html (280 words)

	Reference.com/Encyclopedia/Dedekind domain
		The most important examples of Dedekind domains, and historically the motivating ones, arise from algebraic number fields: start with a finite field extension F of the rational numbers Q and consider the set of all elements of F which are algebraic integers (in other words, the integral closure of Z in F).
		The ideal class group measures the failure of unique factorization in a Dedekind domain (by measuring the failure of ideals to be principal).
		The study of Dedekind domains began when Richard Dedekind introduced the notion of ideal in a ring in the hopes of compensating for the failure of unique factorization into primes in rings of algebraic integers.
	www.reference.com /browse/wiki/Dedekind_domain (594 words)

	AS A EUCLIDEAN DOMAIN; UNIQUE FACTORIZATION THEOREM
		Note that, unlike any abstract Euclidean ring, the quotient and reminder are unique.
		-function behaves under addition, and this was used to prove uniqueness of quotient and remainder; in general, there is no axiom on the
		We repeat the unique factorization theorem, but this time we think specifically of polynomials with coefficients from a field
	www.ma.hw.ac.uk /~takis/teaching_material/infocrypto/Numbers/htmlfiles/node37.html (79 words)

	Unique factorization domain - ExampleProblems.com
		In mathematics, a unique factorization domain (UFD) is, roughly speaking, a commutative ring in which every element can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers.
		A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal.
		An integral domain is a UFD if and only if the ascending chain condition holds for principal ideals, and any two elements of A have a least common multiple.
	www.exampleproblems.com /wiki/index.php/Unique_factorization_domain (557 words)

	Factorization
		In a broad sense, 0 can be factorized, as 01, 02, 03, 05, etc. But none of these are prime factorizations, because 0 is not a prime but is involved.
		The uniqueness is proved by appealing to the characterizing theorem of primes.
		Or, appeal to results in abstract algebra - the set of integers is a Euclidean domain, and therefore a principle ideal domain, and therefore a unique factorization domain.
	www.vex.net /~trebla/numbertheory/factor.html (643 words)

	Chris Brown (Site not responding. Last check: 2007-10-31)
		While it is clear that for any unique factorization domain the concept of GCD is well-defined, it is not always clear how to actually compute GCDs.
		However, many interesting and important unique factorization domains are not Euclidean.
		For example, if D is a unique factorization domain the polynomial ring D[x] is also a unique factorization domain.
	web.usna.navy.mil /~wdj/colloq/talk04_13.html (222 words)

	Math Forum Discussions
		factorization domain, as the example you give shows.
		then when p = 3 it is a unique factorization domain.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.org /kb/thread.jspa?threadID=1090911&messageID=3431508 (349 words)

	Factorization--Unique and Otherwise
		Unique factorization of integers into primes is a fundamental result, and one which goes back to Euclid.
		Definition An integral domain R is a unique factorization domain (UFD) if the conclusion of the Fundamental Theorem of Arithmetic holds for R, i.e., if every non-zero element of R has an essentially unique factorization into irreducibles.
		It is a general theorem that in a UFD, primes and irreducibles are the same, while in a non-UFD, every prime is an irreducible, but an irreducible need not be prime.
	www.ams.org /featurecolumn/archive/factorization.html (914 words)

	ABSTRACT ALGEBRA ON LINE: Unique Factorization
		Let D be a principal ideal domain, and let p be a nonzero element of D. Then p is irreducible in D if and only if pD is a prime ideal of D. Definition.
		Any principal ideal domain is a unique factorization domain.
		Let D be a unique factorization domain, let Q be the quotient field of D, and let f(x) be a primitive polynomial in D[x].
	www.math.niu.edu /~beachy/aaol/unique.html (552 words)

	Integral Domains, Gaussian Integer, Unique Factorization
		And, as a consequence, the prime factorization is unique in Z[i].
		In 1847, G.Lamé (1795-1870) gave a talk at a meeting of the French Academy where he announced a solution to Fermat's Theorem and suggested that the glory of solving the famous theorem should be shared with J.Liouville (1809-1882) to whom he ascribed the method of solution.
		Lamé left his mark in the Theory of Elasticity and Mathematical Physics, he introduced curvilinear coordinates, and in his honor were named an equation, a set of coefficients, and a special class of functions.
	www.cut-the-knot.org /arithmetic/int_domain4.shtml (981 words)

	Relation between gcds in [x] and [x]. Polynomial gcds in non Euclidean domains.
		However we assume that all gcds (for which we have an algorithm or not) are normalized.
		Hence we assume that gcds are uniquely defined.
		Corollary 3 Let R be a UFD with field of fractions K.
	www.csd.uwo.ca /~moreno/CS424/Lectures/ResultantGcd.html/node2.html (575 words)

	Class Log (Site not responding. Last check: 2007-10-31)
		Along the way, we also showed that in a unique factorization domain, the principal ideal generated by any irreducible element is prime.
		In a principal ideal domain, the principal ideal generated by an irreducible element is maximal (and therefore prime).
		The characteristic of an integral domain is either zero or a prime.
	math.bu.edu /people/kea/math542/log.html (602 words)

	Unique factorization domain - Definition, explanation
		Rings which are UFDs are sometimes called factorial, following the terminology of Bourbaki.
		The ring of all complex numbers of the form a + b √ −5, where a and b are integers.
		Most factor ringss of a polynomial ring are not UFDs.
	www.calsky.com /lexikon/en/txt/u/un/unique_factorization_domain.php (576 words)

	Unique Factorization Domain
		Imagine a ring where all irreducible elements are prime.
		Since the factorization of ab is unique, p appears somewhere in the factors of a or b, hence p divides a or b, and p is prime.
		A ring is a ufd iff prime and irreducible elements coincide.
	www.mathreference.com /id,ufd.html (96 words)

	BEACHY/BLAIR: ABSTRACT ALGEBRA
		Among the new additions are a chapter on unique factorization, a number theory thread which runs throughout several optional sections, and an overview of techniques of computing Galois groups.
		In Chapter 3 the abstract definition of a group is introduced, and the students encounter the notion of a group armed with a variety of concrete examples.
		When factor groups are introduced in Chapter 3, we have partitions and equivalence relations at our disposal, and we are able to concentrate on the group structure introduced on the equivalence classes.
	home.att.net /~jabeachy/order.htm (1770 words)

	Unique factorization domains, rings with unique factorization.
		any element of an Unique factorisation domain can be divided in primes elements in essentially only one way.
		We do by 'greatest' mean that they have as many factors in common as possible.
		the unique factorisations of the elements and then combining the common factors.
	hemsidor.torget.se /users/m/mauritz/math/alg/ufact.htm (136 words)

	Jordan Wiens - psifertex.com
		In comparison to steganography, however, cryptography does not involve hiding the information itself, but changing it so that it is rendered incomprehensible to all but the intended parties.
		The is important in cryptography because quantum computers can factor large numbers very, very easily.
		In other words, there's one unique way to factor every number into the primes that compose it.
	psifertex.com /crypt (898 words)

	Math Forum Discussions
		factorization into irreducibles, which in turn follows from the
		For by induction on the number of factors we
		putative counterexample to unique factorization which involves a
	www.mathforum.org /kb/message.jspa?messageID=4117997&tstart=0 (459 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		29(p.400) Let D be a unique factorization domain.
		Let D=Q[X] and let D'=D[Y] be the ring of polynomials with coefficients D. Using results proved in class show that D' is a unique factorization domain.
		Factor X^3-Y^3 into irreducibles in D' and prove that each factor is irreducible.
	www-math.mit.edu /~gyuri/a-h9 (109 words)

	Unique factorization Domain
		Posted: Sat Nov 25, 2006 9:41 pm Post subject: Unique factorization Domain
		a) Prove that I_2, I_3, I'_3 are prime ideals in R. b) Show that the factorizations 6=2.3=(1+sqrt(-5))(1-sqrt(-5))
		Show that these two ideal factorizations give the same factorization of the ideal (6) as the product of prime ideals.
	www.mymathforum.com /viewtopic.php?t=44 (181 words)

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