21. Http//www.math.utsa.edu/ecz/l_a_m.html http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

22. Fundamental Theorem Of Algebra: Information From Answers.com fundamental theorem of algebra In mathematics , the fundamental theorem of algebrastates that every complex polynomial of degree n has exactly n. http://www.answers.com/topic/fundamental-theorem-of-algebra

showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Wikipedia Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping fundamental theorem of algebra Wikipedia fundamental theorem of algebra In mathematics , the fundamental theorem of algebra states that every complex polynomial of degree n has exactly n zeroes , counted with multiplicity. More formally, if (where the coefficients a a n can be real or complex numbers), then there exist ( not necessarily distinct) complex numbers z z n such that This shows that the field of complex numbers , unlike the field of real numbers , is algebraically closed n a and the sum of all the roots equals - a n The theorem had been conjectured in the 17th century but could not be proved since the complex numbers had not yet been firmly grounded. The first rigorous proof was given by Carl Friedrich Gauss in 1799. (An almost complete proof had been given earlier by d'Alembert .) Gauss produced several different proofs throughout his lifetime. All proofs of the fundamental theorem necessarily involve some analysis , or more precisely, the concept of continuity of real or complex polynomials. The main difficulty in the proof is to show that every non-constant polynomial has at least one zero. We mention approaches via

23. PlanetMath: Fundamental Theorem Of Algebra proof of fundamental theorem of algebra (argument principle) (Proof) by rspuzioproof of fundamental theorem of algebra (Rouché s theorem) (Proof) by http://planetmath.org/encyclopedia/FundamentalTheoremOfAlgebra.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About fundamental theorem of algebra (Theorem) Let be a non-constant polynomial . Then there is with In other words, is algebraically closed "fundamental theorem of algebra" is owned by Evandar view preamble View style: HTML with images page images TeX source See Also: complex number complex Attachments: proof of the fundamental theorem of algebra (Liouville's theorem) (Proof) by Evandar proof of fundamental theorem of algebra (Proof) by scanez fundamental theorem of algebra result (Theorem) by rspuzio proof of fundamental theorem of algebra (due to D'Alembert) (Proof) by rspuzio proof of fundamental theorem of algebra (argument principle) (Proof) by rspuzio proof of fundamental theorem of algebra (Rouch©'s theorem) (Proof) by rspuzio Cross-references: algebraically closed polynomial There are 16 references to this object.

24. PlanetMath: Proof Of Fundamental Theorem Of Algebra (Rouché's Theorem) The fundamental theroem of algebra can be proven using Rouché s theorem. proof of fundamental theorem of algebra (Rouché s theorem) is owned by http://planetmath.org/encyclopedia/ProofOfFundamentalTheoremOfAlgebraRouchesTheo

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About proof of fundamental theorem of algebra (Rouch©'s theorem) (Proof) The fundamental theroem of algebra can be proven using simple , it can be thought of as a ``toy model'' (see toy theorem ) for theorems on the zeros of analytic functions . For a variant of this proof in terms of the argument principle fundamental theorem of algebra (argument principle). Proof Denote the order of by Without loss of generality , we may assume that the leading coefficient of is and write Let . Suppose that . Since whenever . Hence, we have the string of inequalities Since polynomials in are analytic functions of in the whole complex plane , they are certainly analytic functions in the disk we conclude that and have the same number of zeros in the disk . Since has a single zero of order at , which counts as zeros according to multiplicity must also have zeros counted according to multiplicity. (By the way we chose

25. Fundamental Theorem Of Algebra - Definition Of Fundamental Theorem Of Algebra In The fundamental theorem of algebra (now considered something of a misnomer bymany mathematicians) states that every complex polynomial of degree n has http://encyclopedia.laborlawtalk.com/Fundamental_theorem_of_algebra

Add to Favorites General Encyclopedia Legal ... Law forum Search Word: Visit our Law forums Labor Law Accidents Law Bankruptcy Law Business Law ... Tax Law The fundamental theorem of algebra (now considered something of a misnomer by many mathematicians) states that every complex polynomial of degree n has exactly n zeroes, counted with multiplicity. More formally, if (where the coefficients a a n can be real or complex numbers), then there exist ( not necessarily distinct) complex numbers z z n such that This shows that the field of complex numbers , unlike the field of real numbers , is algebraically closed n a and the sum of all the roots equals - a n The theorem had been conjectured in the 17th century but could not be proved since the complex numbers had not yet been firmly grounded. The first rigorous proof was given by Carl Friedrich Gauss in 1799. (An almost complete proof had been given earlier by d'Alembert .) Gauss produced several different proofs throughout his lifetime. All proofs of the fundamental theorem necessarily involve some analysis , or more precisely, the concept of continuity of real or complex polynomials. The main difficulty in the proof is to show that every non-constant polynomial has at least one zero. We mention approaches via

26. The Fundamental Theorem Of Algebra (Fine)-Springer Algebra Book The fundamental theorem of algebra states that any complex polynomial must havea complex root. This book examines three pairs of proofs of the theorem from http://www.springeronline.com/sgw/cda/frontpage/0,10735,4-40109-22-1515722-0,00.

Please enable Javascript in your browser to browse this website. Select your subdiscipline Algebra Analysis Applications Mathematical Biology Mathematical Physics Probability Theory Quantitative Finance Home Mathematics Algebra Select a discipline Biomedical Sciences Chemistry Computer Science Economics Education Engineering Environmental Sciences Geography Geosciences Humanities Law Life Sciences Linguistics Materials Mathematics Medicine Philosophy Popular Science Psychology Public Health Social Sciences Statistics preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,4-0-17-900180-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,4-0-17-900170-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,4-0-17-900190-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,4-0-17-900200-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,4-0-17-900369-0,00.gif'); preloadImage('/sgw/cda/pageitems/designobject/cda_displaydesignobject/0,11978,4-0-17-900344-0,00.gif');

27. Fundamental Theorem Of Algebra Internet Resources for the fundamental theorem of algebra Bibliography for thefundamental theorem of algebra short Bibliography for the Fundamental http://math.fullerton.edu/mathews/n2003/FunTheoremAlgebraBib.html

Research Experience for Undergraduates The Fundamental Theorem of Algebra Internet Resources for the Fundamental Theorem of Algebra Bibliography for the Fundamental Theorem of Algebra - short ... Return to Numerical Methods - Numerical Analysis (c) John H. Mathews 2004

28. The Fundamental Theorem Of Algebra An Easy Proof of the fundamental theorem of algebra (in Classroom Notes) CharlesFefferman American Mathematical Monthly, Vol. 74, No. 7. (Aug. http://math.fullerton.edu/mathews/c2003/FunTheoremAlgebraBib/Links/FunTheoremAlg

Bibliography for the Fundamental Theorem of Algebra short Another topological proof of the fundamental theorem of algebra. Elem. Math. 57 (2002), no. 1, 3237, MathSciNet. On the Fundamental Theorem of Algebra Mays J. Lithuanian Mathematical Journal, October 2002, vol. 42, no. 4, pp. 364-372(9), Ingenta. Polynomial interpolation and a multivariate analog of the fundamental theorem of algebra. Hakopian, H. A.; Tonoyan, M. G. East J. Approx. 8 (2002), no. 3, 355379, MathSciNet. A graphical approach to understanding the fundamental theorem of algebra Sudhir Kumar Goel, Denise T. Reid. Mathematics Teacher Dec 2001 v94 i9 p749(1), Expanded Academic. Fundamental theorem of albegra - yet another proof Anindya Sen The American Mathematical Monthly Nov 2000 v107 i9 p842(2), Expanded Academic. A forgotten paper on the fundamental theorem of algebra Frank Smithies Notes and Records Roy. Soc. London 54 (2000), no. 3, 333341, MathSciNet. The fundamental theorem of algebra: a constructive development without choice Fred Richman Pacific J. Math. 196 (2000), no. 1, 213230, MathSciNet.

29. Gauss’s 1799 Proof Of The Fundamental Theorem Of Algebra Teach Gauss’s 1799 Proof Of the fundamental theorem of algebra 66), Gauss’snew proof of the fundamental theorem, written at the age of 21, http://www.21stcenturysciencetech.com/articles/Spring02/Gauss_02.html

EDITORIAL From Spring 2002 21st Century issue. An Induced Mental Block A New Curriculum We have all heard the frequent laments among our co-thinkers and professional colleagues at the sadly reduced state of science and mathematics education in our nation. As in all such matters, after the righteous indignation and hand-wringing, is over, one must ask oneself the realistic question: Are you part of the problem, or part of the solution? If you are not sure, we have a proposal for you. To introduce it, I ask you to perform the following experiment. STEP 1: As a suitable subject, locate any person who has attended high school within the last 50 or so years. You may include yourself. Now, politely ask that person, if he or she would please construct for you a square root. Among the technically educated, it is very common, next, to see the diagonal of the square appear, often with the label 2 attached. As this has nothing whatsoever to do with the solution, I have found it most effective to point out in such cases, that the problem is really much simpler than that. No knowledge of the Pythagorean Theorem, nor any higher mathematics, is required. An Induced Mental Block What is the problem? No student of the classical method of education, which has been around for at least the past 2,500 years, could ever have any problem with this simple exercise. The mental block which arises here is the perfectly lawful result of the absurd and prevalent modern-day teaching that number can exist independent of any physically determining principle. This is the ivory-tower view of mathematics, which holds sway from grade school to university, and reaches up like a hand from the grave, even into the peer review process governing what can be reported as the results of experimental physics.

30. The Fundamental Theorem Of Algebra. The fundamental theorem of algebra. This theorem is also called the theoremof d Alembert. Theorem 6.2.1 Let $ P(z)$ be a non constant polynomial over http://ndp.jct.ac.il/tutorials/complex/node40.html

Next: Weierstrass' Theorem. Up: The theorems of Liouville Previous: Liouville's Theorem. Contents The Fundamental Theorem of Algebra. This theorem is also called the theorem of d'Alembert Theorem 6.2.1 Let be a non constant polynomial over . Then has a root. Corollary 6.2.2 Let be a non constant polynomial of degree over . Then has exactly roots, counted with multiplicity. First examples are displyed in subsection Corollary 6.2.3 Every non constant polynomial with real coefficients is the product of factors of degree 1 and 2. Proof . Let be a polynomial with real coefficients. By thm , this polynomial has at least one root . If this root is real, then factors by Suppose that is not real. By thm is also a root of . Thus, factors by Re An example can be found in Next: Weierstrass' Theorem. Up: The theorems of Liouville Previous: Liouville's Theorem. Contents Noah Dana-Picard 2004-01-26

31. AoPS Math Forum :: View Topic - Fundamental Theorem Of Algebra All times are GMT 7 Hours, fundamental theorem of algebra Post Posted TueApr 12, 2005 646 am Post subject fundamental theorem of algebra http://www.artofproblemsolving.com/Forum/topic-33344.html

Visit the AoPS Book Store Font Size: The time now is Fri Sep 16, 2005 4:31 pm All times are GMT - 7 Hours Fundamental theorem of algebra Mark the topic unread Math Forum College Playground ... Theorems, Formulas and Other Questions Moderators: amfulger fedja Kent Merryfield Moubinool Page of [2 Posts] Printer View Tell a Friend View previous topic View next topic Author Message liyi Navier-Stokes Equations Joined: 17 Jul 2003 Posts: 1534 Posted: Tue Apr 12, 2005 6:46 am Post subject: Fundamental theorem of algebra Theorem Every function has a real or complex root. Sketch of Proof Suppose ) such that where If there is a pair of such that , the theorem is proved. Consider the function then and where is a continuous function of Finally, let us consider and where is positive. If , then the integrand is continuous and it must holds that . Then we prove that the equality fails for large enough. Anyone who likes to complete the proof? Back to top liyi Navier-Stokes Equations Joined: 17 Jul 2003 Posts: 1534 Posted: Tue Apr 12, 2005 7:16 am

32. Fundamental Theorem Of Algebra -- Facts, Info, And Encyclopedia Article fundamental theorem of algebra. Categories Theorems, Field theory, Complexanalysis, Abstract algebra In (A science (or group of related sciences) http://www.absoluteastronomy.com/encyclopedia/f/fu/fundamental_theorem_of_algebr

Fundamental theorem of algebra [Categories: Theorems, Field theory, Complex analysis, Abstract algebra] In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics , the fundamental theorem of algebra states that every complex (A mathematical expression that is the sum of a number of terms) polynomial of degree n has exactly n (Click link for more info and facts about zeroes) zeroes , counted with multiplicity. More formally, if (where the coefficients a a n can be (An old small silver Spanish coin) real or ((psychoanalysis) a combination of emotions and impulses that have been rejected from awareness but still influence a person's behavior) complex numbers), then there exist ( (Click link for more info and facts about not) not necessarily distinct) complex numbers z z n such that This shows that the (A piece of land cleared of trees and usually enclosed) field of (A number of the form a+bi where a and b are real numbers and i is the square root of -1) complex numbers , unlike the field of (Any rational or irrational number) real numbers , is (Click link for more info and facts about algebraically closed) algebraically closed n a and the sum of all the roots equals - a n The theorem had been conjectured in the 17th century but could not be proved since the complex numbers had not yet been firmly grounded. The first rigorous proof was given by

33. Complex Numbers: The Fundamental Theorem Of Algebra Also, the part of the fundamental theorem of algebra which stated there actuallyare n solutions of an nth degree equation was yet to be proved, pending, http://www.clarku.edu/~djoyce/complex/fta.html

Dave's Short Course on The Fundamental Theorem of Algebra As remarked before, in the 16th century Cardano noted that the sum of the three solutions to a cubic equation x bx cx d b , the negation of the coefficient of x . 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". His formulation, which he didn't prove, also gives a general relation between the n solutions to an n th degree equation and its n coefficients. An n th degree equation can be written in modern notation as x n a x n a n x a n x a n where the coefficients a a n a n , and a n are all constants. Girard said that an n th degree equation admits of n solutions, if you allow all roots and count roots with multiplicity. So, for example, the equation x x x + 1 = has the two solutions 1 and 1. Girard wasn't particularly clear what form his solutions were to have, just that there be n of them: x x x n , and x n Girard gave the relation between the n roots x x x n , and x n and the n coefficients a a n a n , and a n that extends Cardano's remark. First, the sum of the roots

34. The Fundamental Theorem Of Algebra (from Algebra) --  Encyclopædia Britannica The fundamental theorem of algebra (from algebra) Descartes s work was the startof the transformation of polynomials into an autonomous object of intrinsic http://www.britannica.com/eb/article-231072

Home Browse Newsletters Store ... Subscribe Already a member? Log in Content Related to this Topic This Article's Table of Contents Expand all Collapse all Introduction Emergence of formal equations ... Problem solving in Egypt and Babylon Greece and the limits of geometric expression The Pythagoreans and Euclid Diophantus The equation in India and China Islamic contributions ... Analytic geometry The fundamental theorem of algebra Impasse with radical methods Galois theory Galois's work on permutations Acceptance of Galois theory ... Number theory Fundamental concepts of modern algebra Prime factorization Fields Ideals Systems of equations ... New challenges and perspectives Additional Reading General history Ancient and Greek algebra Indian and Chinese algebra Islamic algebra ... Print this Table of Contents Shopping Price: USD $1495 Revised, updated, and still unrivaled. The Official Scrabble Players Dictionary (Hardcover) Price: USD $15.95 The Scrabble player's bible on sale! Save 30%.

35. Fundamental Theorem Of Algebra --  Britannica Concise Encyclopedia - The Onlin fundamental theorem of algebra body Theorem of equations proved by Carl FriedrichGauss in 1799. http://www.britannica.com/ebc/article-9354980

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36. Fundamental Theorem Of Algebra@Everything2.com Theorem (fundamental theorem of algebra) Let p be a nonconstant polynomial with The fundamental theorem of algebra (unlike that of calculus) is usually http://www.everything2.com/index.pl?node=fundamental theorem of algebra

37. Fundamental Theorem Of Algebra The fundamental theorem of algebra (FTA) states. Every polynomial of degree nwith complex coefficients has n roots in the complex numbers. http://www.und.nodak.edu/dept/math/history/fundalg.htm

Fundamental Theorem of Algebra The fundamental theorem of algebra (FTA) states Every polynomial of degree n with complex coefficients has n roots in the complex numbers. There are many other equivalent versions of this, for example that every real polynomial can be expressed as the product of real linear and real quadratic factors. Early work with equations only considered positive real roots so the FTA was not relevant. Cardan realized that one could work with numbers outside of the reals while studying a formula for the roots of a cubic equation. While solving x = 15x + 4 using the formula he got an answer involving the square root of -121. He manipulated this to obtain the correct answer, x = 4, even though he did not understand exactly what he was doing with these "complex numbers." In 1572 Bombelli created rules for these "complex numbers." In 1637 Descartes said that one can "imagine" for every equation of degree n n roots, but these imagined roots do not correspond to any real quantity. Albert Girard , a Flemish mathematiciam, was the first to claim that there are always n solutions to a polynomial of degree n in 1629 in . He does not say that the solutions are of the form a + b i , a, b real. Many mathematicians accepted Girard's claim that a polynomial equation must have

38. Fundamental Theorem Of Algebra The fundamental theorem of algebra (FTA) states. Every polynomial of degree nwith complex coefficients has n roots in the complex numbers. http://www.und.nodak.edu/instruct/lgeller/fundalg.html

Fundamental Theorem of Algebra The fundamental theorem of algebra (FTA) states: Every polynomial of degree n with complex coefficients has n roots in the complex numbers. There are many other equivalent versions of this, for example that every real polynomial can be expressed as the product of real linear and real quadratic factors. Early work with equations only considered positive real roots so the FTA was not relevant. Cardan realized that one could work with numbers outside of the reals while studying a formula for the roots of a cubic equation. While solving x = 15x + 4 using the formula he got an answer involving the square root of -121. He manipulated this to obtain the correct answer, x = 4, even though he did not understand exactly what he was doing with these "complex numbers." In 1572 Bombelli created rules for these "complex numbers." In 1637 Descartes said that one can "imagine" for every equation of degree n n roots, but these imagined roots do not correspond to any real quantity. Albert Girard , a Flemish mathematiciam, was the first to claim that there are always n solutions to a polynomial of degree n in 1629 in . He does not say that the solutions are of the form a + b i , a, b real. Many mathematicians accepted Girard's claim that a polynomial equation must have

39. Schiller Institute -Pedagogy - Gauss's Fundamental Theorem Of A;gebra fundamental theorem of algebra. His Declaration of Independence. by Bruce DirectorApril, 2002. To List of Pedagogical Articles To Diagrams Page http://www.schillerinstitute.org/educ/pedagogy/gauss_fund_bmd0402.html

Home Search About Fidelio ... Dialogue of Cultures SCHILLER INSTITUTE Carl Gauss's Fundamental Theorem of Algebra H is Declaration of Independence by Bruce Director April, 2002 To List of Pedagogical Articles To Diagrams Page To Part II Lyndon LaRouche on the Importance of This Pedagogy ... Related Articles Carl Gauss's Fundamental Theorem of Algebra Disquisitiones Arithmeticae Nevertheless, he took the opportunity to produce a virtual declaration of independence from the stifling world of deductive mathematics, in the form of a written thesis submitted to the faculty of the University of Helmstedt, on a new proof of the fundamental theorem of algebra. Within months, he was granted his doctorate without even having to appear for oral examination. 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. d'Alembert, Bougainville, Euler, de Foncenex, Lagrange, and the encyclopedists ... which latter, however, will probably not be much pleased), besides many and varied comments on the shallowness which is so dominant in our present-day mathematics." In essence, Gauss was defending, and extending, a principle that goes back to Plato, in which only physical action, not arbitrary assumptions, defines our notion of magnitude. Like Plato, Gauss recognized it were insufficient to simply state his discovery, unless it were combined with a polemical attack on the Aristotelean falsehoods that had become so popular among his contemporaries.

40. Schiller Institute -Pedagogy - Gauss's Fundamental Theorem Of Alegebra-2 fundamental theorem of algebra. Part II. Bringing the Invisible to the Surface Gauss, in his proofs of the fundamental theorem of algebra, http://www.schillerinstitute.org/educ/pedagogy/gauss_fund_part2.html

Home Search About Fidelio ... Dialogue of Cultures SCHILLER INSTITUTE Carl Gauss's Fundamental Theorem of Algebra Part II Bringing the Invisible to the Surface by Bruce Director May, 2002 To Diagram Page Back to Part I When Carl Friedrich Gauss, writing to his former classmate Wolfgang Bolyai in 1798, criticized the state of contemporary mathematics for its "shallowness", he was speaking literally - and, not only about his time, but also of ours. Then, as now, it had become popular for the academics to ignore, and even ridicule, any effort to search for universal physical principles, restricting the province of scientific inquiry to the, seemingly more practical task, of describing only what's on the surface. Ironically, as Gauss demonstrated in his 1799 doctoral dissertation on the fundamental theorem of algebra, what's on the surface, is revealed only if one knows, what's underneath. Gauss' method was an ancient one, made famous in Plato's metaphor of the cave, and given new potency by Johannes Kepler's application of Nicholas of Cusa's method of On Learned Ignorance. For them, the task of the scientist was to bring into view, the underlying physical principles, that could not be viewed directly-the unseen that guided the seen. Take the illustrative case of Pierre de Fermat's discovery of the principle, that refracted light follows the path of least time , instead of the path of least distance followed by reflected light. The principle of least-distance, is a principle that lies on the surface, and can be demonstrated in the visible domain. On the other hand, the principle of least-time, exists "behind", so to speak, the visible, brought into view, only in the mind. On further reflection, it is clear, that the principle of least-time, was there all along, controlling, invisibly, the principle of least-distance. In Plato's terms of reference, the principle of least-time is of a "higher power", than the principle of least-distance.

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