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Topic: Irreducible element

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Maximal ideal

Glossary of ring theory

Friedrich Nietzsche

	PlanetMath: join irreducibility
		An element is meet irreducible iff it is covered by at most one other element.
		If a lattice satisfies the descending chain condition, then every element can be expressed as a join of join irreducible elements.
		This is version 4 of join irreducibility, born on 2006-03-20, modified 2006-03-25.
	planetmath.org /encyclopedia/Irreducible5.html (179 words)

	element - Definitions from Dictionary.com
		Synonyms: These nouns denote one of the individual parts of which a composite entity is made up: the grammatical elements of a sentence; jealousy, a component of his character; melody and harmony, two of the constituents of a musical composition; ambition as a key factor in her success; humor, an effective ingredient of a speech.
		An element is composed of atoms that have the same atomic number, that is, each atom has the same number of protons in its nucleus as all other atoms of that element.
		Eighty-one of the elements have isotopes that are stable.
	dictionary.reference.com /search?q=element (1559 words)

	PlanetMath: irreducible
		See Also: UFD, divisibility in rings, PID, irreducible polynomial
		This is version 3 of irreducible, born on 2001-11-04, modified 2005-05-01.
		Object id is 668, canonical name is Irreducible.
	planetmath.org /encyclopedia/Irreducible.html (28 words)

	Integral domain
		If a and b are elements of the integral domain R, we say that a divides b or a is a divisor of b or b is a multiple of a if and only if there exists an element x in R such that ax = b.
		The elements which divide 1 are called the units of R; these are precisely the invertible elements in R.
		Every prime element is irreducible (here, for the first time, we need R to be an integral domain), but the converse is only true in a unique factorization domain.
	www.ebroadcast.com.au /lookup/encyclopedia/pr/Prime_element.html (621 words)

	Irreducible
		In Abstract algebra, irreducible is an abbreviation of irreducible element.
		The notions of irreducibility in algebra and manifold theory are related.
		An irreducible manifold is thus prime, although the converse does not hold.
	www.ebroadcast.com.au /lookup/encyclopedia/ir/Irreducible.html (197 words)

	Irreducible polynomial - Wikipedia, the free encyclopedia
		It is helpful to compare irreducible polynomials to prime numbers: prime numbers (together with the corresponding negative numbers of equal modulus) are the irreducible integers.
		Hence, all irreducible polynomials are of degree 1.
		The existence of irreducible polynomials of degree greater than one (without zeros in the original field) historically motivated the extension of that original number field so that even these polynomials can be reduced into linear factors: from rational numbers to real numbers and further to complex numbers.
	en.wikipedia.org /wiki/Irreducible_polynomial (786 words)

	Subfields and extension fields
		Collecting our thoughts, we observe that in, there are primitive elements, and that all non-zero elements of the field can be constructed as powers of the primitive element.
		While any irreducible polynomial can be used to construct the extension field, computation in the field is easier of a primitive polynomial is used.
		All the nonzero elements of the field can be generated as powers of the roots of the primitive polynomial.
	www.engineering.usu.edu /classes/ece/7670/lecture3/node5.html (1081 words)

	Creation Functions
		Given the ring of class functions R of a finite group G with k conjugacy classes and k elements a_i contained in some common cyclotomic field, create a class function on G for which the value on the i-th class is equal to the i-th term a_i.
		Given a ring of class functions R create its zero element (which is the class function that takes on the value 0 on every element of the group).
		The known irreducible characters are stored in the character ring of G. The functions in this section return character tables, which are enumerated sequences of characters that are flagged to allow printing in a special format.
	www.umich.edu /~gpcc/scs/magma/text980.htm (657 words)

	[No title]
		If for example your ring is a PID (principal ideal domain), that's all elements of the domain, as you pointed out, but for number fields non-PID's always have prime ideals which are not principal, so there will be elements in the domain not in Prod_1.
		The factors y and z cannot both have factorizations into irreducibles, so let x_{i+1} be a factor of x_i which also doesn't have a factorization into irreducibles, and which isn't an associate of x_i.
		The nonzero principal ideals (which are generated by nonzero elements, with two elements generating the same ideal if and only if they differ by units) are associated with the identity element of the class group.
	www.math.niu.edu /~rusin/known-math/99/prodprime (902 words)

	Anonymous Comments on Math 114
		If the question asks you to prove that t^4+t^3+t^2+t+1 is irreducible, it would be against the spirit of the problem to answer simply that this is a special case of a polynomial that was treated in class.
		So if the extension over R is finite and the element adjoined (or one of them) is not in R, then the extension is algebraic which means the element must be in C which means the elements adjoined are in C which implies that the degree is 2.
		In a field F with q elements, the reason that x^q = x is that F\{0} is a cyclical multiplicative group with q-1 elements, and x^(q-1)=1.
	math.berkeley.edu /~ribet/114/comments.html (22857 words)

	Rings
		A highest common factor of two elements x,y in a ring A is an element d which is divisible by all those elements of A which divide both x and y.
		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 element x in an integral domain is prime if, for any product yz in the ring, if x divides yz then either x divides y or x divides z.
	mcraefamily.com /mathhelp/BasicAARings.htm (1006 words)

	Galois Fields (Site not responding. Last check: 2007-10-08)
		elements which are residue classes of integers modulo the prime number
		It can be proved that the ring of polynomials over any finite field has at least one irreducible polynomial of every degree.
		as the field and an irreducible polynomial, for what stated in Theorem 16, a new field can be set.
	www.science.unitn.it /~flego/links/tesi1/node15.html (317 words)

	Finite Fields and Polynomials over them
		It is also equal to the order of the subgroup generated by the element.
		Take an arbitrary element and multiply it by itself several times.
		Apply the Frobenius automorphism to its alleged irreducible factor.
	www.podval.org /~sds/finfield.html (666 words)

	Irreducible (mathematics) - Wikipedia, the free encyclopedia
		In abstract algebra, irreducible can be an abbreviation for irreducible element; for example an irreducible polynomial.
		In commutative algebra, a commutative ring R is irreducible if its prime spectrum, that is, the topological space Spec R, is an irreducible topological space.
		A directed graph is irreducible if, given any two vertices, there exists a path from the first vertex to the second.
	en.wikipedia.org /wiki/Irreducible_(mathematics) (399 words)

	ABSTRACT ALGEBRA ON LINE: Unique Factorization (Site not responding. Last check: 2007-10-08)
		Let a and b be elements of a commutative ring R with 1.
		Let a and b be elements of a commutative ring R with identity.
		If a and b are nonzero elements of D, then D contains a greatest common divisor of a and b, of the form as+bt for s,t in D. Furthermore, any two greatest common divisors of a and b are associates.
	www.mat.niu.edu /~beachy/aaol/unique.html (552 words)

	Math 793
		Use the previous to construct an element that has one factorization with precisely two irreducible factors and another with precisely 12 irreducible factors.
		that factors into two irreducible elements and an example of a prime that is not prime in
		Show that if a prime factors into two irreducible elements, then the two elements are primes of
	www.ndsu.nodak.edu /ndsu/coykenda/M793.4.SU2001.htm (67 words)

	Class Log (Site not responding. Last check: 2007-10-08)
		Along the way, we also showed that in a unique factorization domain, the principal ideal generated by any irreducible element is prime.
		The principal ideal generated by a prime element is always prime.
		In a principal ideal domain, the principal ideal generated by an irreducible element is maximal (and therefore prime).
	math.bu.edu /people/kea/math542/log.html (602 words)

	THE WORLD PHENOMENOLOGY INSTITUTE World Institute for Advanced Phenomenological Research
		We discover the footholds of order within the virtualities of the human individual rather than in the abstract structures of the structurizing intellect or in the ideal "essences." While differentiating himself from "other" beings and things, the functioning of the individual being follows the irreducible guidelines of the human condition.
		We may account intentionally for the ethical choice; yet it is not the ethical action which is the pivotal human function, but the creative act which as the prototype of all action, establishes the specificity of man over all other types of beings.
		In this dialogue phenomenology functions not as laying the ground of fixed principles or standards, but as conducting a clarifying scrutiny of the praxis itself.- a universal praxiology of knowledge.
	www.phenomenology.org /article.html (2776 words)

	ABSTRACT ALGEBRA ON LINE: Ideal Theory of Commutative Rings
		K in which each element is a root of a nonzero polynomial with coefficients in K is said to be algebraic over K. There is a similar concept for ring extensions T
		I is the intersection of all prime ideals of R that contain I. In any principal ideal domain, our next definitions both reduce to the statement that the ideal in question is generated by a power of an irreducible element.
		One important consequence of the generalized principal ideal theorem is that any Noetherian ring satisfies the descending chain condition for prime ideals.
	www.math.niu.edu /~beachy/aaol/commutative.html (2296 words)

	Ring Theory
		Prove that in a commutative ring, prime elements are irreducible.
		Give examples of a noncommutative ring with zero divisors, a noncommutative division ring, and integral domain, a UFD, a PID, a Euclidean domain and examples which show that ID Be sure to justify that your examples have or do not have the requisite properties.
		This is the converse of a well-known theorem.
	math.dartmouth.edu /graduate-students/syllabi/sample-questions/algebra/node3.html (274 words)

	Fantasy - Theory
		Fantasy is “a fiction evoking wonder and containing a substantial and irreducible element of the supernatural with which the mortal characters in the story or the readers become on at least partly familiar terms” (Manlove 1).
		In Gothic or dream fictions, the supernatural elements are a “symbolic extension of the purely human mind.”(Manlove) “as soon as it clearly emerges, the science of psychology also begins to color fantasy” (Mathews 26).
		Mathews identifies two complementary strains in modern fantasy, one essentially religious in nature (exemplified by Tolkien and, presumably, by Lewis) and one with a more secular orientation (ex., William Morris) (86).
	www.northern.edu /hastingw/fantasy.htm (521 words)

	Unique factorization domain - Wikipedia, the free encyclopedia
		In R[X,Y,Z,W], then, the degree one element (rα − 1)X + rβY + rγZ + rδW must be an element of the ideal (XY − ZW), but the non-zero elements of that ideal are degree two and higher.
		Next, the element XY equals the element ZW because of the relation XY − ZW = 0.
		That means that XY and ZW are two different factorizations of the same element into irreducibles, so R[X,Y,Z,W] / (XY − ZW) is not a UFD.
	en.wikipedia.org /wiki/Unique_factorization_domain (805 words)

	Part 3: The alienation of labour is the premise for its emancipation \| International Communist Current
		He shows that alienation is an irreducible element of the wage labour system, which can only mean that the more the worker produces, the more he enriches not himself, but capital, this alien power standing over him.
		But under conditions of alienation this advance becomes a disadvance: man's capacity to separate subject from object, which is a fundamental element in the specifically human consciousness, is perverted into a relation of hostility to nature, to the sensuous 'objective' world.
		And yet - and this element is developed in the pages of the Grundrisse in particular - man's history since tribal times can be seen as the continuous dissolution of the original communal bonds which held the first human societies together.
	en.internationalism.org /ir/070_commy_03 (6944 words)

	38a
		The maximal order of any element in that group can be made small, so it would not be unusual for
		then all elements of the set are reducible, since any one irreducible element will have Frobenius which is a full
		with a primitive and irreducible factor of degree
	www.aimath.org /WWN/primesinp/articles/html/38a (1538 words)

	Find in a Library: The teleologies in Husserlian phenomenology : the irreducible element in man, part III, "telos" as ...
		Find in a Library: The teleologies in Husserlian phenomenology : the irreducible element in man, part III, "telos" as the pivotal factor of contextual phenomenology ; papers read at the VIth International Phenomenology Conference, University of Arezzo/Siena, July 1-July 6, 1976
		The teleologies in Husserlian phenomenology : the irreducible element in man, part III, "telos" as the pivotal factor of contextual phenomenology ; papers read at the VIth International Phenomenology Conference, University of Arezzo/Siena, July 1-July 6, 1976
		WorldCat is provided by OCLC Online Computer Library Center, Inc. on behalf of its member libraries.
	www.worldcatlibraries.org /wcpa/ow/e6de21659b452616.html (121 words)

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