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Topic: Modern algebra

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	Algebra - MSN Encarta
		Algebra is a branch of mathematics concerning the study of structure, relation and quantity.
		Algebra is frequently used to solve not just a single equation with a single unknown but also several equations involving multiple unknowns at the same time.
		An important development in algebra in the 16th century was the introduction of modern symbols for unknowns, algebraic powers, and algebraic operations.
	encarta.msn.com /encyclopedia_761552816_5/Algebra.html (1263 words)

	algebra. The Columbia Encyclopedia, Sixth Edition. 2001-07
		Algebra is a generalization of arithmetic and gains much of its power from dealing symbolically with elements and operations (such as addition and multiplication) and relationships (such as equality) connecting the elements.
		Much of classical algebra is concerned with finding solutions to equations or systems of equations, i.e., finding the roots, or values of the unknowns, that upon substitution into the original equation will make it a numerical identity.
		Among the important concepts of modern algebra are those of a matrix and of a vector space.
	www.bartleby.com /65/al/algebra.html (560 words)

	On Algebra - Part1
		Algebra is a branch of mathematics in which symbols, usually letters of the alphabet, represent numbers or members of a specified set and are used to represent quantities and to express general relationships that hold for all members of the set.
		Historically, algebra is the study of solutions of one or several algebraic equations, involving the polynomial (an expression of two or more terms) functions of one or several variables.
		Modern algebraists have increasingly abstracted and axiomatized the structures and patterns of argument encountered not only in the theory of equations, but in mathematics generally.
	www.irfi.org /articles/articles_251_300/on_algebra.htm (1503 words)

	algebra square root solver
		Algebra may divided into "classical algebra" (equation solving or "find the unknown number" problems) and "abstract algebra", also called "modern algebra" (the study of groups, rings, and fields).
		The development of algebraic notation progressed through three stages: the rhetorical (or verbal) stage, the syncopated stage (in which abbreviated words were used), and the symbolic stage with which we are all familiar.
		This advance freed algebra from the consideration of particular equations and thus allowed a great increase in generality and opened the possibility for studying the relationship between the coefficients of an equation an the roots of the equation ("theory of equations").
	softmath.com /tutorials/algebra-square-root-solver.html (2056 words)

	Kurosh: Lectures on general algebra Introduction
		How great, and sometimes decisive, the impact of this modern algebra was on the development of many domains of mathematics, among which we mention, in the first instance, topology and functional analysis, is common knowledge.
		One should have thought that the fundamental ideas and the most important results accumulated in present-day general algebra ought to be part of the scientific equipment of every well-educated mathematician to the same extent as in the thirties, when the majority of candidates in mathematics were examined in modern algebra.
		The object of the book would be to exhibit the main branches of modern general algebra, preferably in their mutual interconnection, the exposition being restricted to individual important theorems and aiming straight at these theorems.
	www-history.mcs.st-andrews.ac.uk /Extras/Kurosh_algebra.html (1428 words)

	Analysis > Early Modern Conceptions of Analysis (Stanford Encyclopedia of Philosophy)
		Algebra was specifically called an ‘art of analysis’, and it was in the work of Descartes (as well as Fermat) that its enormous potential was realised.
		The further application of algebraic techniques, in the context of the development of function theory, was to lead to the creation by Leibniz and Newton of the differential and integral calculus—which, in mathematics, came to be called ‘analysis’.
		Well versed in both classical and modern thought, at the forefront of both mathematics and philosophy, he provided a grand synthesis of existing conceptions of analysis and at the same time paved the way for the dominance of the decompositional conception.
	plato.stanford.edu /entries/analysis/s4.html (2237 words)

	Algebra \| Macmillan Mathematics
		Algebra is a branch of mathematics that uses variables to solve equations.
		An important development in algebra was the introduction of symbols for the unknown in the sixteenth century.
		Although algebra may be something not everyone enjoys, it is one branch of mathematics that is impossible to ignore.
	www.bookrags.com /research/algebra-mmat-01 (684 words)

	Kurosh's Introduction to "Lectures on general algebra"
		How great, and sometimes decisive, the impact of this modern algebra was on the development of many domains of mathematics, among which we mention, in the first instance, topology and functional analysis, is common knowledge.
		One should have thought that the fundamental ideas and the most important results accumulated in present-day general algebra ought to be part of the scientific equipment of every well-educated mathematician to the same extent as in the thirties, when the majority of candidates in mathematics were examined in modern algebra.
		The object of the book would be to exhibit the main branches of modern general algebra, preferably in their mutual interconnection, the exposition being restricted to individual important theorems and aiming straight at these theorems.
	www-groups.dcs.st-and.ac.uk /~history/Extras/Kurosh_algebra.html (1428 words)

	History of Algebra
		Classical" algebra dealt with "concrete objects": real or complex numbers, polynomials with complex coefficients or specific groups of transformations.
		"Modern" algebra replaced these concrete objects by elements of a set, whose nature is irrelevant and whose relationships to each other are specified by axioms.
		This new "abstract" algebra studied sets endowed with one or more operations whose properties are deduced from axioms.
	library.thinkquest.org /C0110248/algebra/history3.htm (90 words)

	Math (Site not responding. Last check: )
		Algebra 2 is a full-year elective course for students interested in increasing their proficiency in mathematics.
		Since algebra is the basic foundation for many careers, students will find this course complements topics covered in Modern Algebra and Algebra I. Graphing calculators will be used throughout the year.
		Modern Algebra is a pre-requisite because the material studied in Modern Algebra 2 is a continuation of the basic algebraic fundamentals learned and practiced in beginning algebra.
	www.lemars.k12.ia.us /math1.htm (991 words)

	Mathematics Algebra
		In the past decades, this branch of algebra has been strongly linked to combinatorics and geometry through seminal work of Auslander and Kac, among others, showing that much of the desired structural information on the nonlinear objects and their linear snapshots is encoded in directed graphs.
		The algebraic side of this field includes the ongoing development of `noncommutative counterparts' to classical algebraic geometry, such as `quantized' versions of the algebras of polynomial functions on algebraic spaces.
		Algebraic Geometry is the study of the solutions of polynomial equations.
	math.ucsb.edu /department/algebra.php (435 words)

	Algebra Help - The History of Algebra
		Algebra provides a generalization of arithmetic by using symbols, usually letters, to represent numbers.
		However, it was not until the 3rd century that algebraic problems began to be considered in a form similar to those studied today.
		The algebraic statements can then be simplified according to the rules of the algebra, and translated into a simpler circuit design.
	www.helpalgebra.com /info/algebrahistory.htm (1214 words)

	The Origins of Modern Algebra
		But rather this was algebra in more or less the sense we use the word today (but without thinking of it in abstract terms), namely the study of structures in which one could work in very much the same way that traditional algebra operates in the realm of rational numbers, real numbers, or complex numbers.
		The other main thread leading to modern commutative ring theory came from algebraic geometry, and I won't really discuss that here except to mention that mathematicians were becoming very aware that the algebra of functions defined on an algebraic curve or surface had a great deal in common with algebraic number rings.
		During this time, the development of Lie groups and algebras (which are non-associative) was proceeding and some of the fundamental concepts in the theory of associative algebras (the concept of the radical, for instance) were developed first for Lie algebras.
	www.math.hawaii.edu /~lee/algebra/history.html (3349 words)

	Modern Algebra (Site not responding. Last check: )
		Since the freeing of algebra from the necessity of geometric interpretation, the letters in an algebraic expression had continued to be considered as standing for numbers, and the symbols between the letters were for the operations of ordinary arithmetic.
		A further outcome of the freeing of algebra from arithmetic was the study of algebraic structures which had the properties not possessed by the algebra of numbers.
		Examples were the algebra of quaternions, introduced by William Rowan Hamilton (1805-65) in 1844, and the algebra of the "Laws of Thought" (now called Boolean Algebra), introduced by George Boole (1815-64) in 1854.
	scitsc.wlv.ac.uk /university/scit/modules/mm2217/ma.htm (552 words)

	Free online textbooks, videos, tutorials, lecture notes,
		Algebra and Analysis for Computer Science, by Jean Gallier.
		Abstract Algebra with GAP, Authored by By J. Rainbolt and J. Gallian.
		Elements of Abstract and Linear Algebra by Edwin H. Connell.
	homepages.nyu.edu /~jmg336/html/mathematics.html (5019 words)

	Modern Algebra
		Over the years, the phrase "abstract algebra" has overtaken the less accurate "modern algebra" --- the subject is now more retro than neo.
		Modern Algebra at FSU is distinct from the course Intro to Abstract Algebra, which is taken by all mathematics majors.
		The goals are similar, but Modern Algebra targets the math education major instead of the math major.
	www.math.fsu.edu /~huckaba/modernalgSp04.html (336 words)

	Open Directory - Science:Math:Algebra (Site not responding. Last check: )
		Please submit abstract or modern algebra websites to the main category and linear and high school algebra webpages to their appropriate subcategories.
		Algebra is a branch of mathematics that uses letters or other symbols to represent unknown quantities, called variables.
		Algebra tutorials that are geared towards students should be placed in Science/Math/High_School and not in the Algebra Education category.
	dmoz.org /Science/Math/Algebra/desc.html (338 words)

	MTH 233: Modern Algebra
		Modern algebra rests on a foundation of set theory and logic.
		We can allow the field of scalars to be the field of rational numbers (so important in number theory), the field of complex numbers, the field of rational functions (complex analysis and differential equations), or even a finite field (coding theory).
		The simplest ring is the ring of integers (1 divided by 2 is not an integer), but examples of interesting rings range from the finite rings of modular (clock) arithmetic to the ring of matrices and the ring of linear differential operators studied at Smith in the physical chemistry course.
	math.smith.edu /Local/guide/node34.html (412 words)

	Modern Algebra 1: Overview
		Just as regular algebra allowed you to perform operations in which the exact nature of the numbers is left unknown, so in modern algebra we allow the exact nature of the operations to be unknown.
		Any algebraic system that satisfies these axioms is called a vector space, and there are many different sorts of vector spaces, made up of matrices, or columns of numbers, or polynomials, or functions, to name a few.
		The second is a feasible goal for a modern algebra course, although we may not be able to reach it before the end of the year.
	www.american.edu /academic.depts/cas/mathstat/People/kalman/modalg/overview.html (642 words)

	Applications of modern algebra
		I thought that abstract algebra was the same thing as "modern" algebra.
		A Linear Algebra course (matrices) would be more useful, since you use matrix math some to solve large simultaneous equation problems in EE, and since SPICE is based on matrix math a fair amount.
		Groups and algebras are used extensively in elementary particle physics and quantum field theory.
	www.physicsforums.com /showthread.php?t=115272 (1085 words)

	OUP: UK General Catalogue
		The history, however, of what is recognized today as high school algebra is much shorter, extending back to the sixteenth century, while the history of what practicing mathematicians call "modern algebra" is even shorter still.
		In addition to considering the technical development of various aspects of algebraic thought, the historians of modern algebra whose work is united in this volume explore such themes as the changing aims and organization of the subject as well as the often complex lines of mathematical communication within and across national boundaries.
		Among the specific algebraic ideas considered are the concept of divisibility and the introduction of non-commutative algebras into the study of number theory and the emergence of algebraic geometry in the twentieth century.
	www.oup.com /uk/catalogue/?ci=9780821843437 (335 words)

	Read This: Algebra in Ancient and Modern Times
		The author gives a complete and concise treatment of the controversies surrounding the solution of the cubic equation, and also shows how it was the solution to the cubic equation, not the solution of the quadratic equation, that led to serious study of complex numbers.
		The final section, "Some themes from modern algebra" is a glimpse into the Pandora's box of algebraic creatures unleashed following the development of complex numbers in the 1500's.
		I would particularly recommend Algebra in Ancient and Modern Times to strong high school students, to high school algebra teachers, to people who want a history of mathematics with a lot of mathematics in the history, and to anyone who needs to know how to find an analytic solution to a nasty fourth degree polynomial.
	www.maa.org /reviews/aamt.html (643 words)

	MATH W4041 Introduction to Modern Algebra I (Site not responding. Last check: )
		Math 1202 (Multivariable Calculus) and Math 2010 (Linear Algebra), or equivalent courses, are prerequisites for this course.
		You should also be familiar with complex numbers, mathematical induction and other methods of proof, and in general have a certain confidence in your abilities to handle abstract mathematical reasoning.
		There are very many texts in Abstract Algebra; browsing the library is recommended for further examples, history, or different treatment of the material.
	www.math.columbia.edu /~rf/modernalg1.html (530 words)

	Math 121: Modern Algebra II
		If L is an algebraic extension of K and if K is an algebraic extension of F, then K is an algebraic extension of F. The composite K_1_2 of two subfields K_1 and K_2.
		A complex number r is an algebraic integer if and only if Z[r] (under addition) is contained in a finitely generated subgroup of C. The algebraic integers form a ring.
		An algebraic integer that is rational is an integer.
	math.stanford.edu /~white/121_1084/121_1084.htm (1205 words)

	Gene Expression: The invention of algebra & the modern mind
		Our modern world has been characterized by a progressive ratcheting up of the minimum mathematical competencies demanded of individuals, and as this occurs it is expected that eventually one will encounter a piont at where a large portion of humanity simply can not be reached by practically implementable didactic methods.
		The main flaw with Cohen's argument that algebra is not necessary in "real life" anyway is that it would have to be extended to all forms of formal reasoning.
		While modern algebraic symbols are a relatively new invention, that doesn't mean that algebraic notation is difficult.
	scienceblogs.com /gnxp/2006/02/the_invention_of_algebra_the_m.php (1603 words)

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