41. The Fundamental Theorem Of Algebra. How to think of a proof of the fundamental theorem of algebra. Prerequisites.A familiarity with polynomials and with basic real analysis. http://www.dpmms.cam.ac.uk/~wtg10/ftalg.html

How to think of a proof of the fundamental theorem of algebra Prerequisites A familiarity with polynomials and with basic real analysis. Statement Every polynomial (with arbitrary complex coefficients) has a root in the complex plane. (Hence, by the factor theorem, the number of roots of a polynomial, up to multiplicity, equals its degree.) Preamble How to come up with a proof. If you have heard of the impossibility of solving the quintic by radicals, or if you have simply tried and failed to solve such equations, then you will understand that it is unlikely that algebra alone will help us to find a solution of an arbitrary polynomial equation. In fact, what does it mean to solve a polynomial equation? When we `solve' quadratics, what we actually do is reduce the problem to solving quadratics of the particularly simple form x =C. In other words, our achievement is relative: if it is possible to find square roots, then it is possible to solve arbitrary quadratic equations. But is it possible to find square roots? Algebra cannot help us here. (What it can do is tell us that the existence of square roots does not lead to a contradiction of the field axioms. We simply "adjoin" square roots to the rational numbers and go ahead and do calculations with them - just as we adjoin i to the reals without worrying about its existence. See my

42. About "The Fundamental Theorem Of Algebra" , The fundamental theorem of algebra states that any complex polynomialmust have a complex root. This basic result, whose first accepted proof...... http://mathforum.org/library/view/11467.html

The Fundamental Theorem of Algebra Library Home Full Table of Contents Suggest a Link Library Help Visit this site: http://www.springeronline.com/sgw/cda/frontpage/0,10735,4-40109-22-1515722-0,00.html Author: B. Fine, Fairfield Univ., CT; G. Rosenberger, Univ. of Dortmund, Germany Description: The Fundamental Theorem of Algebra states that any complex polynomial must have a complex root. This basic result, whose first accepted proof was given by Gauss, lies at the intersection of the theory of numbers and the theory of equations, and arises also in many other areas of mathematics. This book examines three pairs of proofs of the theorem from three different areas of mathematics: abstract algebra, complex analysis and topology. Levels: College Languages: English Resource Types: Textbooks Math Topics: Modern Algebra Complex Analysis Algebraic Number Theory Algebraic Topology ... Help http://mathforum.org/

43. About "Fundamental Theorem Of Algebra" fundamental theorem of algebra. _ LibraryHome Full Table of Contents Suggest a Link Library Help http://mathforum.org/library/view/11461.html

Fundamental Theorem of Algebra Library Home Full Table of Contents Suggest a Link Library Help Visit this site: http://www.cut-the-knot.org/do_you_know/fundamental.html Author: Interactive Mathematics Miscellany and Puzzles, Alexander Bogomolny Description: Some of the history of complex numbers, perfect numbers, irrational numbers, imaginary numbers, and the first proof of the Fundamental Theorem of Algebra (statement and significance), given by Carl Friedrich Gauss (1777-1855) in his Ph.D. Thesis (1799). Levels: High School (9-12) Early College Languages: English Math Topics: Equations Imaginary/Complex Numbers Home The Math Library ... Help http://mathforum.org/

44. American Mathematical Monthly, The: Fundamental Theorem Of Algebra And Linear Al Full text of the article, fundamental theorem of algebra and linear algebra,The from American Mathematical Monthly, The, a publication in the field of http://www.findarticles.com/p/articles/mi_qa3742/is_200308/ai_n9300786

@import url(/css/us/style1.css); @import url(/css/us/searchResult1.css); @import url(/css/us/articles.css); @import url(/css/us/artHome1.css); Home Advanced Search IN free articles only all articles this publication Automotive Sports FindArticles American Mathematical Monthly, The Aug/Sep 2003 Content provided in partnership with 10,000,000 articles Not found on any other search engine. Featured Titles for ASA News ASEE Prism Academe African American Review ... View all titles in this topic Hot New Articles by Topic Automotive Sports Top Articles Ever by Topic Automotive Sports fundamental theorem of algebra and linear algebra, The American Mathematical Monthly, The Aug/Sep 2003 by Derksen, Harm Save a personal copy of this article and quickly find it again with Furl.net. It's free! Save it. 1. INTRODUCTION. The first widely accepted proof of the fundamental theorem of algebra was published by Gauss in 1799 in his Ph.D. thesis, although by today's standards this proof has gaps. In 1814 Argand gave a proof (with only small gaps) that was based on a flawed 1746 proof of d'Alembert. Many other proofs followed, including three more by Gauss. For more about the history of the fundamental theorem of algebra, see [5] or [6]. Proofs of the fundamental theorem of algebra can be divided roughly into three categories (see [3] for a collection of proofs). First there are the topological proofs (see [1] or [8]). These proofs are based on topological considerations such as the winding number of a curve in R^sup 2^ around 0. Gauss's original proof might fit under this heading as well. Then there are analytical proofs (see [9]), which are related to Liouville's theorem: a nonconstant entire function on C is unbounded. Finally, there are the algebraic proofs (see [4] or [10]). These proofs use only the fact that every polynomial of odd degree with real coefficients has a real root and that every complex number has a square root. The deeper reasons why these arguments work can be understood in terms of Galois theory.

45. The Fundamental Theorem Of Algebra - Chapter 6 Review: Section 7 It s the fundamental theorem of algebra. It s no big surprise, though, as Mrs.Gould taught it to us during Section 4. The theorem simply states that the http://webpages.charter.net/thejacowskis/chapter6/section7.html

Chapter 6 Review: Section 7 - Using the Fundamental Theorem of Algebra Notes The Fundamental Theorem of Algebra Tada! It's the Fundamental Theorem of Algebra . It's no big surprise, though, as Mrs. Gould taught it to us during Section 4 . 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. Finding the Zeroes of Polynomial Functions Once again, this is not new material. Now, before you use the zero product property, make sure that you factor out complex numbers, too. Writing Polynomial Functions with Given Factors We've been factoring polynomial functions for the whole chapter, but how do the textbook writers make polynomial functions that factor? First, it is necessary to understand that anything with a factor of a square root has at least two factors, one positive and one negative. Since i is the square root of -1, anything that has i for a factor also of - i With that in mind, writing polynomial functions with given factors is easy. Just multiply all the factors together. The product is the polynomial of least degree and a with a leading coefficient of 1 that has all the factors. Practice Quiz Find all the zeroes of the function.

46. Read About Fundamental Theorem Of Algebra At WorldVillage Encyclopedia. Research fundamental theorem of algebra. Everything you wanted to know about Fundamentaltheorem of algebra but had no clue how to find it.. Learn about Fundamental http://encyclopedia.worldvillage.com/s/b/Fundamental_theorem_of_algebra

Culture Geography History Life ... WorldVillage Fundamental theorem of algebra From Wikipedia, the free encyclopedia. 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 complex analysis topology , and algebra Find a closed disk D of radius r p z p z r p z D is therefore achieved at some point z in the interior of D p z m p z ) is a holomorphic function in the entire complex plane. Applying

47. Merak MML Browsing fundamental theorem of algebra. POLYNOM575 theorem for b1 being Functionlikequasi_total finite-Support Relation of NAT,the carrier of F_Complex http://merak.pb.bialystok.pl/mmlquery/fillin.php?entry=POLYNOM5:75&comment=Funda

48. A Graphical Approach To Understanding The Fundamental Theorem Of Algebra—MT A Graphical Approach to Understanding the fundamental theorem of algebra Algebra/Algebraic Thinking, Graphing Functions, Technology, Calculators http://my.nctm.org/eresources/article_summary.asp?URI=MT2001-12-749a&from=B

49. Gauss's Second Proof Of The Fundamental Theorem Of Algebra Gauss s second proof of the fundamental theorem of algebra. Another new proof ofthe theorem that every integral rational algebraic function of one variable http://www.cs.man.ac.uk/~pt/misc/gauss-web.html

Gauss's second proof of the fundamental theorem of algebra Another new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree HTML DVI compressed PostScript A5 PS booklet ... PDF by Carl Friedrich Gauss (1815); the Latin original appears in Volume 3, pages 33-56, of his collected works. Any polynomial of even degree m is transformed into one of degree m m 1); notice that, although this is typically a larger number, it contains one fewer factor of 2. Each root of the derived polynomial determines a pair of roots of the original one via a quadratic equation. Any odd-degree equation has a real root. This English translation was made by Paul Taylor in December 1983 and corrected by Bernard Leak. A summary of the proof, together with a note by Martin Hyland on its logical significance, appeared in Eureka (1985). The L

50. Historia Matematica Mailing List Archive: Re: [HM] The Fundamental Theorem Of Al Arnold, BH A topological proof of the fundamental theorem of algebra. Amer.Math. Monthly 56 (1949), 465 466. This proof is based on the http://sunsite.utk.edu/math_archives/.http/hypermail/historia/nov98/0055.html

Re: [HM] The Fundamental Theorem of Algebra Gerhard Warnecke warnecke@gmd.de Thu, 5 Nov 1998 17:12:51 +0100 Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Next message: Roger Cooke: "Re: [HM] Is Greek mathematics the *real* thing?" Previous message: Valdusek@aol.com: "Re: [HM] Is Greek mathematics the *real* thing?" In reply to: Samuel S. Kutler: "[HM] The Fundamental Theorem of Algebra" Next in thread: Julio Gonzalez Cabillon: "Re: [HM] The Fundamental Theorem of Algebra" Next in thread: Sanford L. Segal: "Re: [HM] The Fundamental Theorem of Algebra" I am responding to the e-mails communicated by Samuel S. Kutler, Martin Davis, Eric Detrez and Gordon Fischer on 3. 11. 1998. Mr. Kutler asked some questions, namely 1 Number of proofs, Gauss gave? 2 Rigor of Gauss' proofs? 3 The role of topology in the proof of the fundamental theorem of algebra (FTA)? while the other writers commented on the third question which indeed caused Koerper" as the (German speaking) creators of this theory called it) These questions can all be answered by the relevant literature.

51. Fundamental Theorem Of Algebra - Linix Encyclopedia In mathematics, the fundamental theorem of algebra states that every complexpolynomial of degree n has exactly n zeroes, counted with multiplicity. http://web.linix.ca/pedia/index.php/Fundamental_theorem_of_algebra

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 complex analysis topology , and algebra Find a closed disk D of radius r p z p z r p z D is therefore achieved at some point z in the interior of D p z m p z ) is a holomorphic function in the entire complex plane. Applying

52. FundamentalTheoremOfAlgebra fundamental theorem of algebra (English). Search for Fundamental theorem ofalgebra in NRICH PLUS maths.org Google http://thesaurus.maths.org/mmkb/entry.html?action=entryByConcept&id=713&langcode

53. Homotopies And Fundamental Theorem Of Algebra And More homotopies and fundamental theorem of algebra and more by Amy (April 2, 2003).Re f no fixpoint deg(f)=1 by NN. (April 3, 2003) http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2003;task=show_msg;msg=01

54. Fundamental Theorem Of Algebra Thread. fundamental theorem of algebra by Matty (April 13, 2005). Re FundamentalTheorem of Algebra by Henno Brandsma (April 13, 2005) http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2005;task=show_msg;msg=24

55. FTA Project The fundamental theorem of algebra Project. In the group of Henk Barendregt,a number of people have coded the full proof of a significant mathematical http://www.cs.ru.nl/~freek/fta/

The "Fundamental Theorem of Algebra" Project In the group of Henk Barendregt , a number of people have coded the full proof of a significant mathematical theorem in the computer. 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. This page briefly presents the project. Five people have contributed to the coding: Herman Geuvers Freek Wiedijk Jan Zwanenburg Randy Pollack ... Henk Barendregt Herman is the person who started the project and who manages it. Apart from Randy, these people all work in Nijmegen. Randy contributed remotely from Edinburgh, keeping contact by e-mail and CVS. The type theory of Coq naturally corresponds to a constructive logic, so it was decided to translate a constructive proof of the Fundamental Theorem. 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. We decided to treat the real numbers axiomatically, as a "parameter" to the development (because constructive real numbers were needed, the axiomatic real numbers from the Coq distribution weren't usable and an own version of the real number axioms was created). Because of this approach, any representation of the constructive real numbers can be "plugged in" into the proof.

56. The Fundamental Theorem Of Algebra Math reference, the fundamental theorem of algebra. http://www.mathreference.com/at,fta.html

Algebraic Topology, The Fundamental Theorem of Algebra Search Site map Contact us Main Page algebraic Topology Use the arrows at the bottom to step through Algebraic Topology. The Fundamental Theorem of Algebra A homotopic proof that an n degree polynomial has a root proceeds by contradiction. Suppose p(z) is a monic, n degree polynomial with no roots. The term z n dominates, for large enough z, so there exists some radius r such that every z beyond r has p(z) nonzero. For instance, r can be set to the sum of the norms of the coefficients. Since p is monic, r is at least 1. to S Apply a linear homotopy that shrinks all the coefficients, except for the lead coefficient, down to 0. We are left with the function z n , which wraps the circle around itself n times. Therefore our original function f has degree n. previous section , we showed that a function of degree n cannot be extended continuously to a disk, thus we have a contradiction, and p(z) has a root after all. when a root w is found, divide p(z) by z-w to give a lower degree polynomial. Repeat this process until you have n roots, some of which may be duplicates. Roots Inside the Unit Circle to S . We will show that the degree is the number of roots inside the unit circle. Split p(z) into n linear factors z-w i . This splits f into the product of z-w i i i out to z, as z traces the unit circle. For instance, if w

57. Fundamental Theorem Of Algebra Math reference, the fundamental theorem of algebra. http://www.mathreference.com/cx,fta.html

Complex Numbers, Fundamental Theorem of Algebra Search Site map Contact us Download this Site Main Page Complex Numbers Use the arrows at the bottom to step through Complex Numbers. Fundamental Theorem of Algebra The complex numbers form a closed field What?! Let's put it another way. Take any polynomial p(z), where the coefficients are complex numbers. There is some complex number r that is a root of p(z). In other words, some r satisfies p(r) = 0. Divide through by z-r and find the next root, and so on, until p is the product of monomials z-r. This can be done for every polynomial p(z). This is the fundamental theorem of algebra. There is a beautiful proof using Galois theory , but for those familiar with analytic functions, Liouville's theorem does the trick. Note that p(z) is dominated by its leading term. If p(z) has degree 4, then z dominates everything for large enough z, even if the coefficient on z is small. As z approaches infinity, far from the origin, p(z) approaches infinity. Every nontrivial polynomial has a root in the complex numbers.

58. Fundamental Theorem Of Algebra fundamental theorem of algebra. fundamental theorem of algebra. An $n\mathrm{th}$ degree polynomial has $ n$ (not necessarily distinct) zeros. http://aah.ryan-usa.com/node45.html

Next: Rational Zero Theorem Up: Algebra Previous: Factors and Zeros of Fundamental Theorem of Algebra An degree polynomial has (not necessarily distinct) zeros. Although all of these zeros may be imaginary, a real polynomial of odd degree must have at least one real zero. aah@ryan-usa.com

59. Fundamental Theorem Of Algebra Complex P''(''z Numbers Real fundamental theorem of algebra Complex P ( z Numbers Real Economy. http://www.economicexpert.com/a/Fundamental:theorem:of:algebra.htm

var GLB_RIS='http://www.economicexpert.com';var GLB_RIR='/cincshared/external';var GLB_MMS='http://www.economicexpert.com';var GLB_MIR='/site/image';GLB_MML='/'; document.write(''); document.write(''); document.write(''); document.write(''); A1('s',':','html'); Non User A B C ... Home First Prev [ 1 Next Last 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

60. Chronology Of The Life Of Carl F. Gauss Dissertation contains first proof of the fundamental theorem of algebra.Later proofs in 1815, 1816, 1849. 1800, January, recieves Legendre s essay on the http://www.geocities.com/RainForest/Vines/2977/gauss/appendix/chrono.html

Chronology of the Life of Carl F. Gauss Year Major event(s) April 30, born in Brunswick Enters St. Katherine's School in Brunswick Enters Büttner arithmetic class. Büttner ordered him a textbook from Hamburg Friendship with Bartels. They study the binomial theorem and infinite series together. Bartels leaves the Büttner school. Gauss enters second class of "gynamsium." Exhibits great talent in languages. Presented at court to the Duke of Brunswick. Minister of state Geheimrat Feonçe von Rotenkreuz presents him a table of logarithms. February 18, enters the Collegium Carolinium, supported by the Duke of Brunswick. Perfects himself in ancient and modern languages. Studies the works of Newton, Euler, and Lagrange. march, discovered by induction the fundamental theorem for quadratic resideues (already published by Legendre in 1795). October 11, leaves Brunswick. October 15, registers as student in the University of Göttingen. Application of his methods of least squares. March 30, discovers inscription of the regular polygon of seventeen sides in a circle. April 8, proof that -1 is the quadratic residue of all primes of the form 4n + 1 and nonresidue of the fundamental theorem for quadratic residues.

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