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Fundamental Theorem Of Algebra:     more detail

The Fundamental Theorem of Algebra (Undergraduate Texts in Mathematics) by Benjamin Fine, Gerhard Rosenberger, 1997-06-20 Constructive aspects of the fundamental theorem of algebra;: Proceedings of a symposium conducted at the IBM Research Laboratory, Zurich-Ruschlikon, Switzerland, June 5-7, 1967, Constructive Aspects of the Fundamental Theorem of Algebra by Bruno & Peter Henrici. Eds. Dejon, 1969 Constructive Aspects of the Fundamental Theorem of Algebra by Bruno and Peter Henrici. Eds. Dejon, 1969 Descent: An optimization point of view on different fields [An article from: European Journal of Operational Research] by J. Brinkhuis, 2007-08-16

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1. Fund Theorem Of Algebra
   The fundamental theorem of algebra (FTA) states. Every polynomial equation ofdegree n with complex coefficients has n roots in the complex numbers.  
   http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fund_theorem_of_algebra.h
   
   Extractions: Version for printing The Fundamental Theorem of Algebra (FTA) states Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers. In fact there are many equivalent formulations: for example that every real polynomial can be expressed as the product of real linear and real quadratic factors. Early studies of equations by al-Khwarizmi (c 800) only allowed positive real roots and the FTA was not relevant. Cardan was the first to realise that one could work with quantities more general than the real numbers. This discovery was made in the course of studying a formula which gave the roots of a cubic equation. The formula when applied to the equation x x + 4 gave an answer involving -121 yet Cardan knew that the equation had x = 4 as a solution. He was able to manipulate with his 'complex numbers' to obtain the right answer yet he in no way understood his own mathematics. Bombelli , in his Algebra , published in 1572, was to produce a proper set of rules for manipulating these 'complex numbers'. Descartes in 1637 says that one can 'imagine' for every equation of degree n n roots but these imagined roots do not correspond to any real quantity.

2. Gauss
   One of the alltime greats, Gauss began to show his mathematical brilliance at the early age of seven. He is usually credited with the first proof of The fundamental theorem of algebra.  
   http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Gauss.html
   
   Extractions: Version for printing At the age of seven, Carl Friedrich Gauss In 1788 Gauss began his education at the Gymnasium binomial theorem and the arithmetic- geometric mean, as well as the law of quadratic reciprocity and the prime number theorem. , whom Gauss often ridiculed. His only known friend amongst the students was Farkas Bolyai . They met in 1799 and corresponded with each other for many years. ruler and compasses This was the most major advance in this field since the time of Greek mathematics and was published as Section VII of Gauss's famous work, Disquisitiones Arithmeticae Gauss returned to Brunswick where he received a degree in 1799. After the Duke of Brunswick had agreed to continue Gauss's stipend, he requested that Gauss submit a doctoral dissertation to the University of Helmstedt. He already knew Pfaff , who was chosen to be his advisor. Gauss's dissertation was a discussion of the

3. The Fundamental Theorem Of Algebra
   The fundamental theorem of algebra  
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4. Fund Theorem Of Algebra References
   J Pla i Carrera, The fundamental theorem of algebra before Carl Friedrich Gauss,Publ. Mat. 36 (2B) (1992), 879911. A Fryant and VLN Sarma, Gauss first  
   http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Printref/Fund_theorem_of_
   
   Extractions: J Pla i Carrera, The fundamental theorem of algebra before Carl Friedrich Gauss, Publ. Mat. A Fryant and V L N Sarma, Gauss' first proof of the fundamental theorem of algebra, Math. Student Archive for History of Exact Sciences R C F Kooistra, Gauss and the fundamental theorem of algebra (Dutch), Nieuw Tijdschr. Wisk. S S Petrova, From the history of the analytic proofs of the fundamental theorem of algebra (Russian), History and methodology of the natural sciences XIV : Mathematics, mechanics (Moscow, 1973), 167-172. S S Petrova, The first proof of the fundamental theorem of algebra (Bulgarian), Fiz.-Mat. Spis. B'lgar. Akad. Nauk. I Schneider, Herausragende Einzelleistungen im Zusammenhang mit der Kreisteilungsgleichung, dem Fundamentalsatz der Algebra und der Reihenkonvergenz, in Carl Friedrich Gauss (1777-1855) (Munich, 1981), 37-63. B L van der Waerden, A History of Algebra (Berlin, 1985). JOC/EFR May 1996 MacTutor History of Mathematics

5. S.O.S. Math - Algebra
   Finding Roots of Polynomials Graphically and Numerically. The fundamental theorem of algebra. Factoring Some Special Cases  
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6. Fundamental Theorem Of Algebra -- From MathWorld
   Courant, R. and Robbins, H. The fundamental theorem of algebra. §2.5.4 in WhatIs Mathematics? An Elementary Approach to Ideas and Methods, 2nd ed.  
   http://mathworld.wolfram.com/FundamentalTheoremofAlgebra.html
   
   Extractions: MATHWORLD - IN PRINT Order book from Amazon Algebra Polynomials Fundamental Theorem of Algebra Every polynomial equation having complex coefficients and degree has at least one complex root . This theorem was first proven by Gauss . It is equivalent to the statement that a polynomial of degree has values (some of them possibly degenerate) for which . Such values are called polynomial roots . An example of a polynomial with a single root of multiplicity is , which has as a root of multiplicity 2. SEE ALSO: Degenerate Frivolous Theorem of Arithmetic Polynomial Polynomial Factorization ... [Pages Linking Here] REFERENCES: Courant, R. and Robbins, H. "The Fundamental Theorem of Algebra." §2.5.4 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 101-103, 1996. Krantz, S. G. "The Fundamental Theorem of Algebra." §1.1.7 and 3.1.4 in

7. The Fundamental Theorem Of Algebra
   The fundamental theorem of algebra If P(z) is a polynomial of degree n, then P has at least one zero.  
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8. Fundamental Theorem Of Algebra
   fundamental theorem of algebra. Complex numbers are in a sense perfect while The fundamental theorem of algebra establishes this reason and is the topic  
   http://www.cut-the-knot.org/do_you_know/fundamental.shtml
   
   Extractions: Sites for parents Complex numbers are in a sense perfect while there is little doubt that perfect numbers are complex. Starting from the tail, perfect numbers have been studied by the Ancients ( Elements, IX.36 ). Euler (1707-1783) established the form of even perfect numbers. [Conway and Guy, p137] say this: Are there any other perfect numbers? ... All we know about the odd ones is that they must have at least 300 decimal digits and many factors. There probably aren't any! Every one would agree it's rather a complex matter to write down a number in excess of 300 digits. Allowing for a pun, if there are odd perfect numbers they may legitimately be called complex. What about complex numbers in the customary sense? There is at least one good reason to judge them perfect. The Fundamental Theorem of Algebra establishes this reason and is the topic of the discussion below. In the beginning there was counting which gave rise to the natural numbers (or integers ): 1,2,3, and so on. In the space of a few thousand years, the number system kept getting expanded to include fractions, irrational numbers, negative numbers and zero, and eventually complex numbers. Even a cursory glance at the terminology would suggest (except for fractions) the reluctance with which the new numbers have been admitted into the family.

9. Fundamental Theorem Of Algebra
   fundamental theorem of algebra Gauss' Proof of the fundamental theorem of algebra Translated by Ernest Fandreyer, M.S., Ed.D. Professor  
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10. Fundamental Theorem Of Algebra
    The fundamental theorem of algebra is a theorem about equation solving. It statesthat every polynomial equation over the field of complex numbers of degree  
    http://www.cut-the-knot.org/do_you_know/fundamental2.shtml
    
    Extractions: Sites for parents We already discussed the history of the development of the concept of a number. Here I would like to undertake a more formal approach. Thus, in the beginning there was counting. But soon enough people got concerned with equation solving. (If I saw 13 winters and my tribe's law allows a maiden to marry after her 15th winter, how many winters should I wait before being allowed to marry the gorgeous hunter who lives on the other side of the mountain?) The Fundamental Theorem of Algebra is a theorem about equation solving. It states that every polynomial equation over the field of complex numbers of degree higher than 1 has a complex solution. Polynomial equations are in the form P(x) = a n x n + a n-1 x n-1 + ... + a x + a where a n is assumed non-zero (for why to mention it otherwise?), in which case n is called the degree of the polynomial P and of the equation above. a i 's are known coefficients while x is an unknown number. A number a is a solution to the equation P(x) = if substituting a for x renders it

11. Fundamental Theorem Of Algebra
    The applet on this page is designed for experimenting with the fundamental theorem of algebra, which state that all polynomials with complex  
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12. Fundamental Theorem Of Algebra - Wikipedia, The Free Encyclopedia
    In mathematics, the fundamental theorem of algebra states that every complexpolynomial of degree n has exactly n roots (zeroes), counted with multiplicity.  
    http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra
    
    Extractions: In mathematics , the fundamental theorem of algebra states that every complex polynomial of degree n has exactly n roots (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 . An easy consequence is that the product of all the roots equals (−1) 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

13. Complex Numbers
    Solution of quadratics, solution of cubics 2. The fundamental theorem of algebra 3. The number i. The fundamental theorem of algebra proved!  
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14. En.wikipedia.org/wiki/Fundamental_Theorem_of_Algebra
    fundamental theorem of algebraThe applet on this page is designed for experimenting with the fundamental theoremof algebra, which state that all polynomials with complex coefficients  
    http://en.wikipedia.org/wiki/Fundamental_Theorem_of_Algebra

15. ABSTRACT ALGEBRA ON LINE Contents
    Fitting's lemma for modules(10.4.5) Frattini's argument(7.8.5) fundamental theorem of algebra(8.3.10) Fundamental theorem of arithmetic(1.2.6)  
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16. Fundamental Theorem Of Algebra
    fundamental theorem of algebra. fundamental theorem of algebra.\fbox{\emph{Every $n$thorder polynomial possesses exactly  
    http://ccrma.stanford.edu/~jos/mdft/Fundamental_Theorem_Algebra.html

17. MA 109 College Algebra Notes
    Exercises. Chapter 3 Polynomial Equations. Polynomials. Rational Roots. The fundamental theorem of algebra. Cubic Equations. Quartic  
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18. Complex Numbers : Fundamental Theorem Of Algebra
    fundamental theorem of algebra. Let P (z) = be a polynomial of degree n (withreal or complex coefficients). The fundamental theorem of algebra states  
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19. Lee Lady A Graduate Course In Algebra
    of algebra, viz. classification theorems. We see this ideal actually achieved in only a few other places in algebra the Fundamental Theorem of  
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20. Mathwords: Fundamental Theorem Of Algebra
    fundamental theorem of algebra A polynomial p(x) = anxn + an–1xn–1 + ··· + a2x2+ a1x + a0 with degree n at least 1 and with coefficients that may be real  
    http://www.mathwords.com/f/fundamental_thm_algebra.htm

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