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Topic: Field of quotients

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	Field (mathematics) - Wikipedia, the free encyclopedia
		In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.
		Fields are important objects of study in algebra since they provide a useful generalization of many number systems, such as the rational numbers, real numbers, and complex numbers.
		For a given field F, the set F(X) of rational functions in the variable X with coefficients in F is a field; this is defined as the set of quotients of polynomials with coefficients in F.
	en.wikipedia.org /wiki/Field_(mathematics) (1581 words)

	Fractional ideal - Wikipedia, the free encyclopedia
		The definition of fractional ideal relies on the definitions of module and field of quotients.
		It is an integral domain, and so it possesses a field of quotients K.
		The quotient group of fractional ideals divided by principal fractional ideals is isomorphic to the ideal class group of R.
	en.wikipedia.org /wiki/Fractional_ideal (425 words)

	PlanetMath: field adjunction
		is a field, it must contain all possible quotients of the elements of
		Cross-references: ring, quotients, subring, integral domain, quotient field, contains, subfield, extension field, field
		This is version 9 of field adjunction, born on 2004-05-30, modified 2005-03-28.
	planetmath.org /encyclopedia/FieldAdjunction.html (73 words)

	Adjoined Elements
		The field extension produced by adjoining x to the field f, written f(x), is the smallest field that contains f and x.
		In an earlier section we proved that a finite extension is an integral domain iff it is a field.
		The extension f/p is a field iff p is irreducible.
	www.mathreference.com /fld,adj.html (1149 words)

	APPENDIX J
		The characteristic of a field is its characteristic as a division algebra.
		In this construction, the field elements (or marks in the language of [Dickson 1900]) are residue classes of the integers J modulo the prime p.
		The prime subfield of a finite field is the submodule of the field generated by unity.
	graham.main.nc.us /~bhammel/FCCR/apdxJ.html (5929 words)

	[No title]
		Additionally it should be noted that for a near field situation both electric and magnetic measurement are required (use of E and H sensors).
		dipole, bi-conical, log-periodic etc, and the magnetic component (H) of the electromagnetic field is usually measured with loop sensors (as the current induced in the loop is proportional to the magnetic field strength crossing the loop).
		The values to be assessed (for the electric and magnetic field) are the peak value and “rms” value of the pulsed field.
	www.ero.dk /documentation/docs/doc98/official/Word/REC0204.doc (3448 words)

	620-321 Algebra
		Rings topics include abstract rings and isomorphisms; examples including matrix rings and polynomial rings; homomorphisms, ideals and quotient rings; integral domains and the field of quotients; units, irreducibles and primes; prime and maximal ideals; integral domains and the field of quotients; and Euclidean domains and principal ideal domains.
		Modules topics include submodules, homomorphisms of modules and quotient modules; free modules and bases; structure of a finitely generated module over a principal ideal domain; and applications to abelian groups and to Jordan normal form of matrices.
		Field theory topics include field extensions and their construction; and the degree of a field extension.
	www.unimelb.edu.au /HB/2004/subjects/620-321.html (242 words)

	PlanetMath: classification of covering spaces
		The reason for this is that this theorem provides a direct analogy for the fundamental theorem of Galois theory.
		This theorem provides a correspondence between subgroups of the Galois group of a cover and covers that are its quotients, just as the fundamental theorem of Galois theory provides a correspondence between subextensions of a field extension and subgroups of its Galois group.
		Both fundamental theorems can be viewed as special cases of a more general theorem in the category of schemes; the correct tool is the study of étale morphisms and the étale fundamental group.
	planetmath.org /encyclopedia/ClassificationOfCoveringSpaces.html (306 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		He said, "If F is a field, F[X] is defined as follows: let X be any object (an apple, a cousin, C#, etc.) and form all formal expressions..." But this was fraudulent.
		They are up there with the field extensions, and they have been there since the beginning of time, and those classes are not going to die with you and me, either.
		Of course, in daily life we don't keep the brackets, though it is convenient to use the fractional notation because it reminds us of the applicaton of rational numbers to daily measurement and pizzas.
	www.math.rochester.edu /people/faculty/rarm/plato.html (2608 words)

	ORDINAL REAL NUMBERS 1
		This is the third paper of a series of five papers that have as goal the definition of topological complete linearly ordered fields (continuous numbers) that include the real numbers and are obtained from the ordinal numbers in a method analogous to the way that Cauchy derived the real numbers from the natural numbers.
		The ordinal integers are semigroup-rings of quotient monoids of semigroups that are used to define as semigroup-rings the hierarchy of integral domains of the transfinite integers (see [Gleyzal A. pp 586).I use the term hierarchy not only as a well ordered sequence but also as a net (thus partially ordered).
		It is elementary in algebra that if the integral domain is linearly ordered then also its field of quotients (localization) with the previous definition for its set of positive elements, is a linearly ordered field with the restriction of its ordering on the integral domain to coincide with the ordering of the integral domain.
	www.softlab.ntua.gr /~kyritsis/PapersInMaths/InfinityandStochastics/OR1.htm (3795 words)

	Advanced
		An introduction to proof techniques (including quantifiers and induction), elementary set theory, equivalence relations, and cardinality; followed by an introduction to the topology of the real numbers and elementary real analysis, including rigorous topological and analytic treatments of convergence of sequences and continuity of functions.
		A study of the calculus of vector valued functions and vector fields and an introduction to partial differential equations.
		Topics include normal subgroups, quotient groups, homomorphisms, Cayley's theorem, permutation groups, ideals, the field of quotients of an integral domain, and polynomial rings.
	www.davidson.edu /math/frontpage/advanced_courses.htm (563 words)

	Modern Algebra II Lecture Notes, 09/17/03 (Site not responding. Last check: 2007-11-06)
		In fact we will show how to construct the "smallest" field that has this property, the field of quotients of D. Definition (from Section 0.2): Let S be a nonempty set.
		F contains an isomorphic copy of D -- namely, the subset {[(a,1)], where a lies in D} -- and F is the "smallest" such field in the following sense: any field F' that contains an isomorphic copy of D must have (an isomorphic copy of) F as a subfield of F'.
		Read (5.4) The Field of Quotients of an Integral Domain (again).
	www.assumption.edu /Alfano/MAT352-FA03/Notes/091703.html (442 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Then $R$ is a field if and only if $\{0\}$ and $R$ are the only ideals in $R$.
		Then the quotient $R/I$ is a ring with multiplication \[ (a+I) (b+I) = ab +I. \end{prop} \begin{thm} \textbf{First Isomorphism Theorem} Let $\phi: R \to R'$ be a surjective homomorphism between two rings.
		Any field with these properties is called the \textsl{field of quotients} of $D$.
	www.mcs.drexel.edu /~rboyer/courses/math534/week1.txt (1406 words)

	Articles - P-adic number (Site not responding. Last check: 2007-11-06)
		The point is that p-adic numbers form a field if and only if p is a prime power, and you get the same result for a prime power as you do for the prime (e.g., base 16 is just shorthand for base 2).
		It should be noted that the existence of such a field isomorphism relies on the axiom of choice, and no explicit isomorphism can be given.
		Suppose D is a Dedekind domain and E is its quotient field.
	www.gaple.com /articles/P-adic_numbers (1892 words)

	Baylor University \|\| Baylor Department of Mathematics \|\| Course Descriptions
		Topics include permutation groups, group and ring homomorphisms, direct products of groups and rings, quotient objects, integral domains, field of quotients, polynomial rings, unique factorization domains, extension fields, and finite fields.
		Prerequisite(s): MTH 4314 and consent of the instructor.
		Field theory, Galois theory, modules, finitely generated modules, principal ideal domains, homological methods, and Wedderburn-Artin theorems.
	www.baylor.edu /Math/index.php?id=21292 (970 words)

	Boehmians
		Burzyk and P. Mikusinski, A generalization of the construction of a field of quotients with applications in analysis, Int.
		Mikusinski, Convergence dans l'ensemble des quotients de suites (French), [Convergence in the space of quotients of sequences], C.
		Mikusinski and P. Mikusinski, Quotients de suites et leurs applications dans l'analyse fonctionnelle (French), [Quotients of sequences and their applications in functional analysis], C.
	www.math.ucf.edu /~piotr/boehmians.html (767 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Special emphasis is placed on the concept of isomorphism as well as applications to the algebra of the secondary education classroom.
		Understand how the algebra of the secondary education classroom fits into the field of algebra generally.
		Ring Theory (15 hours) Commutative and non-commutative rings, rings with unity, integral domains, division rings, fields, the field of quotients of an integral domain, maximal and prime ideals, factor rings, and polynomial rings.
	www.lhup.edu /ucc/mathematics/MATH310_rev.doc (533 words)

	ACM Sigplan Notices 28, 11 (Nov 1993), 22-27.
		The level of abstraction in abstract algebra and abstract data types may have been taken too far--we learn about abstract rings and fields instead of enjoying the beauty of particular rings and fields--e.g., number theory.
		Once the basic concepts are in hand, we should revel in the polymorphism of Gaussian integers as pairs of integers, as well as atomic elements of the Gaussian ring, much as we revel in the dual role of C "ints" as both integers and bit strings.
		We know that for a Gaussian integer divisor m+ni, we need m^2+n^2 distinct remainders, so the most elegant choice of representative remainders is a square of area m^2+n^2 which is tilted at an angle of atan(n/m);[5] equivalently, we consider the remainder fraction r/d to reside in an upright square of area 1.
	home.pipeline.com /~hbaker1/Gaussian.html (6038 words)

	Modern Algebra II Lecture Notes, 02/05/03 (Site not responding. Last check: 2007-11-06)
		Also notice that the smallest field that contains the numbers 0 and 1 is the field of rational numbers, Q.
		In the present section we investigate the question of constructing the field of quotients of an integral domain.
		Suppose we are given an integral domain D, and we wish to construct "rational" expressions in D so as to create the smallest field that contains D as a subset.
	www.assumption.edu /Alfano/MAT352-SP03/Notes/020503.html (207 words)

	List of publications
		Burzyk, J.; Mikusinski, P.; A generalization of the construction of a field of quotients with applications in analysis, Int.
		Mikusinski, P., Convergence dans l'ensemble des quotients de suites (French), [Convergence in the space of quotients of sequences], C. Acad.
		Mikusinski, J.; Mikusinski, P., Quotients de suites et leurs applications dans l'analyse fonctionnelle (French), [Quotients of sequences and their applications in functional analysis], C. Acad.
	www.math.ucf.edu /~piotr/publications.html (900 words)

	[No title]
		of relation of divisibility; properties; division algorithm theorem (there is a quotient and reminder) - g.c.d and lcm: def., properties; 2.2 Mathematical induction, recursive definitions, well-ordering 2.3 Prime factorization: Primes, The Fundamental Th.
		field; subfield; examples: Zp; Q,R,C; integral domains (def.; example: polynomials, Z); 4.2 “Completing”/extending an integral domain (from Z to Q= fractions): the field of quotients (def.
		& Th.) 4.3 & 4.6 Q & R: - Q as field of quotients of Z; - def.
	www.ilstu.edu /~lmiones/236rvf03.doc (880 words)

	Dorota Jarosz: Wash U Ph.D. Program :
		Rings: matrix rings, polynomial rings, homomorphism, ideals, maximal and prime ideals, field of quotients, principal ideal domains.
		Fields: characteristic, dimension, geometric constructions, splitting a polynomial, finite fields.
		Fields: Extension of fields, finite fields, Galois theory, Kummer theory.
	webpages.charter.net /djarosz/courses.htm (834 words)

	Michael E. O'Sullivan: Modern Algebra (Site not responding. Last check: 2007-11-06)
		A good understanding of the basics of groups, rings and fields (Math 521A and 521B is enough).
		Integral Domains: ideals, the quotient of an integral domain by an ideal, homomorphisms.
		The field of quotients of an ideal (I will cover this).
	www-rohan.sdsu.edu /~mosulliv/Courses/algebra05s.html (330 words)

	ABSTRACT ALGEBRA: OnLine Study Guide, Section 5.4 (Site not responding. Last check: 2007-11-06)
		The field Q(D) defined in Definitino 5.4.2 is called the field of quotients or field of fractions of D. Theorem 5.4.6.
		Let D be an integral domain that is a subring of a field F. If each element of F has the form ab
		for some a, b in D, the F is isomorphic to the field of quotients Q(D) of D. Forward to §6.1
	www.math.niu.edu /~beachy/abstract_algebra/study_guide/54.html (206 words)

	studychum.com :: Index (Site not responding. Last check: 2007-11-06)
		This forum introduces the basic concepts and recent developments in the fields of dynamical systems and chaos, including stability of equilibria and renormalisation theory of transitions to chaos.
		Students should develop the ability to analyse simple nonlinear discrete and continuous dynamical systems, and to chart parameter regions of stability, periodicity and chaos.
		Groups topics include abstract groups, examples including matrix groups and permutation groups; homomorphisms, normal subgroups, quotients and the first homomorphism theorem; group actions and permutation groups; conjugacy classes and their interpretation in symmetry groups, permutation groups and matrix groups.
	www.studychum.com /phpbb2/index.php?c=49 (2428 words)

	CGAL Support Library Reference Manual: Quotient (Site not responding. Last check: 2007-11-06)
		An object of the class Quotient is an element of the field of quotients of the integral domain type NT.
		A Quotient q is represented as a pair of NTs, representing numerator and denominator.
		There are two access functions, namely to the numerator and the denominator of a quotient.
	www.cs.uu.nl /CGAL/Information/doc_html/ref-manual3/NumberTypeSupport/Quotient.html (169 words)

	GraduateProgram: Math, ASU (Site not responding. Last check: 2007-11-06)
		Group tables, subgroups, cosets, normal subgroups, quotient groups, Lagrange's Theorem, groups of small order, cyclic groups, permutation, alternating, and dihedral groups, simple groups, homomorphisms, isomorphism theorems, products of groups, finitely generated abelian groups, Sylow theorems.
		Ideals, quotient rings, homomorphisms, isomorphism theorems, integral domains, field of quotients, prime and maximal ideals, characteristic, matrix rings, Euclidean rings, polynomial rings, unique factorization theorems, extension fields, degree of an extension, roots of polynomials, finite fields.
		As complete ordered field; inf and sup of a subset; lim inf and lim sup of a sequence and of a function; infinite series, tests for convergence, absolute and conditional convergence.
	math.la.asu.edu /~grad/doc/syllabi.html (1380 words)

	New Page 2 (Site not responding. Last check: 2007-11-06)
		Show that the ring R has characteristic 2 (a + a = 0).
		Show that a finite integral domain is a field.
		Describe the field of quotients of the integral subdomain
	cas.memphis.edu /rfaudree/E3'4261.htm (283 words)

	List of Relevant Courses Taken (Site not responding. Last check: 2007-11-06)
		It demonstrates the power of topological methods in dealing with problems involving shape and position of objects and continuous mappings, and shows how topology can be applied to many areas, including geometry, analysis, group theory and physics.
		Topics include topological spaces and continuous maps; quotient spaces; homotopy and fundamental groups; surfaces; covering spaces; and an introduction to homology theory.
		This page, its contents and style, are the responsibility of the author and do not necessarily represent the views, policies or opinions of The University of Melbourne.
	www.cs.mu.oz.au /~twidjaja/course.html (1202 words)

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