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Topic: Fundamental theorem of algebra

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	PlanetMath: fundamental theorem of algebra
		"fundamental theorem of algebra" is owned by Mathprof.
		proof of fundamental theorem of algebra (argument principle)
		This is version 9 of fundamental theorem of algebra, born on 2002-02-13, modified 2006-12-03.
	planetmath.org /encyclopedia/FundamentalTheoremOfAlgebra.html (85 words)

	Schiller Institute -Pedagogy - Gauss's Fundamental Theorem of Alegebra-2
		Gauss, in his proofs of the fundamental theorem of algebra, showed that even though this transcendental physical principle was outside the visible domain, it nevertheless cast a shadow that could be made visible in what Gauss called the complex domain.
		For example, in the algebraic equation x2 = 4, "x" signifies the side of a square whose area is 4, while, in the equation x2 = -4, the "x" signifies the side of a square whose area is -4, an apparent impossibility.
		In the algebraic equation x3=8, there appears to be only one number, 2 which satisfies the equation, and this number signifies the length of the edge of a cube whose volume is 8.
	www.schillerinstitute.org /educ/pedagogy/gauss_fund_part2.html (2854 words)

	Fund theorem of algebra
		Bombelli, in his Algebra, published in 1572, was to produce a proper set of rules for manipulating these 'complex numbers'.
		In 1814 the Swiss accountant Jean Robert Argand published a proof of the FTA which may be the simplest of all the proofs.
		Euler gave the most algebraic of the proofs of the existence of the roots of an equation, the one which is based on the proposition that every real equation of odd degree has a real root.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Fund_theorem_of_algebra.html (1534 words)

	MAT 200 Lecture Notes -- The Fundamental Theorem of Linear Algebra
		The fundamental theorem of linear algebra tells you the dimension of all four of these subspace if you know the dimension of any one of them.
		[Fundamental Theorem of Linear Algebra] The dimension of the row space of a matrix plus the dimension of the left null space is equal to the number of rows.
		The dimension of the row space, or of the column space-the two spaces have the sme dimension according to the preceding theorem-is called the rank of the matrix.
	www.math.princeton.edu /~stalker/200f99/notes_6.html (1494 words)

	Fundamental - Wikipedia, the free encyclopedia
		Fundamentalism, the belief in, and usually the strict adherence to, the basic or "fundamental" ideas of a system of thought.
		Fundamental theorem of algebra, regarding factorization of polynomials.
		Fundamental theorem of arithmetic, theorem regarding prime factorization.
	en.wikipedia.org /wiki/Fundamental (161 words)

	MA 109 College Algebra Chapter 4
		Intuitively, a function f(x) is said to continuous at b if functional values f(x) are as close as we would like to f(b) as soon as x is sufficiently close to b and at a place where f(x) is defined.
		Theorem 1: (Gauss) Every non-constant polynomial with complex coefficients has at least one complex root.
		Proof of the Fundamental Theorem of Algebra Suppose f(z) is a non-constant polynomial with complex coefficients and no complex roots.
	www.msc.uky.edu /ken/ma109/lectures/fta.htm (3625 words)

	Fundamental Theorem of Algebra (Site not responding. Last check: 2007-09-17)
		D'Alembert in 1746 and Euler in 1749 attempted proofs of the FTA.
		Gauss is credited with the first proof of the FTA in his doctoral thesis of 1799.
		Jean Robert Argand published a proof of the FTA and two years later Gauss published a second proof.
	www.und.nodak.edu /dept/math/history/fundalg.htm (451 words)

	SparkNotes: Polynomials: Introduction and Summary
		The Factor Theorem, which follows from the Remainder Theorem, provides an easy way for determining whether a given binomial is a factor of a given polynomial.
		These are the Conjugate Zeros Theorem and the Fundamental Theorem of Algebra.
		As the name of the theorem implies, polynomial functions and their roots are fundamental to the study of algebra.
	www.sparknotes.com /math/algebra2/polynomials/summary.html (486 words)

	On the Fundamental Theorem of Algebra
		The note that presents a short proof of the Fundamental Theorem of Algebra follows (in an HTML rendition) the message from Professor Vaggione.
		In most traditional textbooks on complex variables, the Fundamental Theorem of Algebra is obtained as a corollary of Liouville's theorem using elementary topological arguments.
		The difficulty presented by such a scheme is that the proofs of Liouville's theorem involve complex integration which makes the reader believe that a proof of the Fundamental Theorem of Algebra is too involved.
	www.cut-the-knot.org /fta/vaggione.shtml (391 words)

	The Fundamental Theorem of Algebra (Site not responding. Last check: 2007-09-17)
		The intermediate value theorem provides a positive square root for every positive real number, and a root to any odd degree polynomial in the reals, as x moves from -∞ to +∞.
		The first sylow theorem provides a subgroup h whose order is a power of 2, while the index of h is odd.
		Since l/R is finite and separable, apply the primitive element theorem and write l = R(u).
	www.mathreference.com /fld-sep,fta.html (515 words)

	Schiller Institute -Pedagogy - Gauss's Fundamental Theorem of A;gebra
		Describing his intention to his former classmate, Wolfgang Bolyai, Gauss wrote, "The title [fundamental theorem] indicates quite definitely the purpose of the essay; only about a third of the whole, nevertheless, is used for this purpose; the remainder contains chiefly the history and a critique of works on the same subject by other mathematicians (viz.
		Girolamo Cardan (1501-1576), and later, Leibniz, showed that there was a "hole" in all forms of algebraic equations, as indicated by the appearance of the square roots of negative numbers, as solutions to such equations.
		Like Leibniz, Gauss rejected the deductive approach of investigating algebraic equations on their own terms, insisting that it was physical action that determined the characteristics of the equations.
	www.schillerinstitute.org /educ/pedagogy/gauss_fund_bmd0402.html (3122 words)

	Complex numbers: the fundamental theorem of algebra
		By the 17th century the theory of equations had developed so far as to allow Girard (1595-1632) to state a principle of algebra, what we call now "the fundamental theorem of algebra".
		Certainly complex numbers of the form a + b√–1 were sufficient to solve quadratic equations, but it wasn't clear they were enough to solve cubic and higher-degree equations.
		Also, the part of the Fundamental Theorem of Algebra which stated there actually are n solutions of an nth degree equation was yet to be proved, pending, of course, some description of the possible forms that the solutions might take.
	www.clarku.edu /~djoyce/complex/fta.html (430 words)

	Fundamental Theorem of Algebra
		The Fundamental Theorem of Algebra establishes this reason and is the topic of the discussion below.
		Leonhard Euler (1707-1783) made complex numbers commonplace and the first proof of the Fundamental Theorem of Algebra was given by Carl Friedrich Gauss (1777-1855) in his Ph.D. Thesis (1799).
		Remarks on Proving The Fundamental Theorem of Algebra
	www.cut-the-knot.org /do_you_know/fundamental.shtml (795 words)

	3.4 - Fundamental Theorem of Algebra
		If you check out fundamental in the dictionary, you will see that it relates to the foundation or the base or is elementary.
		Fundamental theorems are important foundations for the rest of the material to follow.
		Here are some of the fundamental theorems or principles that occur in your text.
	www.richland.edu /james/lecture/m116/polynomials/theorem.html (457 words)

	A Nonstandard Proof of the Fundamental Theorem of Algebra American Mathematical Monthly, The - Find Articles (Site not responding. Last check: 2007-09-17)
		A Nonstandard Proof of the Fundamental Theorem of Algebra
		The "Fundamental Theorem of Algebra" states that the field of complex numbers is algebraically closed-every polynomial f(z) = a^sub o^ + a^sub 1^z +...
		The theorems of real analysis are valid for the hyperreals because the latter satisfy all the axioms used in proving these theorems.
	www.findarticles.com /p/articles/mi_qa3742/is_200510/ai_n15715107 (947 words)

	FTA Project
		The theorem chosen for this project was the "Fundamental Theorem of Algebra" (which states that every non-constant polynomial P over the complex numbers has a "root", i.e., that every non-trivial polynomial equation P(z)=0 always has a solution in the complex plane), and the system used was the Coq system from France.
		The proof that was chosen was the so-called "Kneser" proof, which analyzes an iterative proces that converges to one of the roots of the equation.
		Basic algebraic structures: groups, rings, fields, ordered fields, etc. (because the development is constructive, it is not based on the customary notion of "Setoids", but instead it is based on "CSetoids" in which "apartness" is the fundamental notion).
	www.cs.ru.nl /~freek/fta/index.html (683 words)

	Gauss’s 1799 Proof Of the Fundamental Theorem of Algebra
		66), Gauss’s new proof of the fundamental theorem, written at the age of 21, was an explicit and polemical attack on the shallow misconceptions of his celebrated predecessors.
		As Gauss showed for the case of the solution of algebraic functions, and as was already recognized in the writings of Plato, a higher concept of magnitude requires an act of the mind.
		Let the mastery of Gauss’s fundamental theorem as developed in his revolutionary 1799 proof, serve as a cornerstone of a new curriculum for secondary and university undergraduate students.
	www.21stcenturysciencetech.com /articles/Spring02/Gauss_02.html (1284 words)

	[No title] (Site not responding. Last check: 2007-09-17)
		On the other hand, whenever a polynomial has been factored into only linear and irreducible quadratics, then it has been factored completely, since both linear factors and irreducible quadratics cannot be factored any further over the real numbers.
		High school students' algebra experience should enable them to create and use tabular, symbolic, graphical, and verbal representations and to analyze and understand patterns, relations, and functions with more sophistication than in the middle grades.
		In grades 9—12, students should develop an understanding of the algebraic properties that govern the manipulation of symbols in expressions, equations, and inequalities.
	www.angelfire.com /extreme3/zq15ky/chapman2.html (572 words)

	Fundamental Theorem of Geometry
		--------------------------------------------------- Some theorem designated "The Fundamental Theorem of....." is a bit arbitrary since any of the mathematical disciplines you refer to rests upon a number of definitions, theorems, etc. I suppose for example that the "Fundamental Theorem of Euclidean Geometry" is the parallel line theorem, since that distinguishes plane geometry from convex and concave geometries.
		I think the bottom line is there is nothing "fundamental" about the "fundamental" theorems.
		It just means that those theorems are significant departures from some contrasting prior discipline.
	www.newton.dep.anl.gov /askasci/math99/math99241.htm (174 words)

	The Fundamental Theorem of Algebra - Chapter 6 Review: Section 7
		The theorem simply states that the degree of any polynomial is how many solutions it has.
		Remember though, the solutions may not all be real, and solutions that are real may be irrational.
		Section 7 - Using the Fundamental Theorem of Algebra
	webpages.charter.net /thejacowskis/chapter6/section7.html (249 words)

	Math 1513 College Algebra Fall 2000
		Polynomial and Rational Functions – Remainder theorem, factor theorem, Fundamental Theorem of Algebra, rational zero theorem, approximating zeros, asymptotes of rational functions
		During the first three weeks of the semester, you will need to take an algebra skills pretest and pass at 80% correct.
		Of course, when that happens, the person being cheated is you because you are robbing yourself of an opportunity to learn.
	home.snu.edu /~lturner/CollegeAlgebra/Syllabus.htm (846 words)

	Mathwords: Fundamental Theorem of Algebra
		The theorem that establishes that, using complex numbers, all polynomials can be factored.
		A generalization of the theorem asserts that any polynomial of degree n has exactly n zeros, counting multiplicity.
		Mathwords: Terms and Formulas from Algebra I to Calculus
	www.mathwords.com /f/fundamental_thm_algebra.htm (72 words)

	GeoSci 236: The Fundamental Theorem of Linear Algebra
		GeoSci 236: The Fundamental Theorem of Linear Algebra
		Figure 1: The forward problem (the fundamental theorem of linear algebra).
		This may be justified as the least conjectural, or most parsimonious, solution (given our imperfect information), but it is far from clear that these are the solution's most desirable attributes.
	geosci.uchicago.edu /~gidon/geosci236/fundam (618 words)

	David - Fundamental Theorem of Algebra
		The proof of Taylor's Theorem led me to Rolle's Theorem.
		This led me to the Extreme Value Theorem, the proof of which the book said was beyond the book's scope.
		So, I searched for the Extreme Value Theorem, and that led me to a Note on the Extreme Value Theorem.
	util.crazylife.org /60035.html (91 words)

	hofprints - On the Fundamental Theorem of Algebra (Site not responding. Last check: 2007-09-17)
		Edwards, Harold M. On the Fundamental Theorem of Algebra.
		Everyone knows what is meant by the fundamental theorem of algebra: The field of complex numbers is algebraically closed.
		My goal in this half hour is to convince you that a different theorem deserves the name.
	hofprints.hofstra.edu /31 (111 words)

	Roots of unity; the Fundamental Theorem of Algebra (Site not responding. Last check: 2007-09-17)
		Roots of unity; the Fundamental Theorem of Algebra
		Using complex numbers, we have a more sophisticated version of the Fundamental Theorem of Algebra:
		Then this decomposition is unique up to appropriate permutations of the real factors amongst themselves and of the complex factors amongst themselves.
	www.math.pitt.edu /~sparling/23014/23014notes6/node32.html (178 words)

	History of Mathematics: History of Algebra (Site not responding. Last check: 2007-09-17)
		History of algebraic geometry: an outline of the history and development of algebraic geometry.
		A historical survey of algebraic methods of approximating the roots of numerical higher equations up to the year 1819.
		A history of algebra: from al-Khwarizmi to Emmy Noether.
	aleph0.clarku.edu /~djoyce/mathhist/algebra.html (114 words)

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