fundamental theorem of algebra proof

The fundamental theorem of algebra has quite a few number of proofs (enough to fill a book!). Like many ﬁrst courses in Linear Algebra, we could easily be content with just accepting the statement of the theorem and deferring a discussion of its proof to a more advanced mathematics course. There is a short proof of the fundamental theorem of algebra based upon Liouville's theorem. In fact, it seems a new tool in mathematics can prove its worth by being able to prove the fundamental theorem in a different way. The Tropical Semiring 2 3. A consequence of the theorem is that "genuinely different" entire functions cannot dominate each other, i.e. The diversity of proof techniques available is yet another indication of how fundamental and deep the Fundamental Theorem of Algebra really is. No entire function dominates another entire function. There is a very simple, short and elegant proof of the Fundamental Theorem of Algebra based on Liouville’s Theorem. University of Hyderabad Hyderabad 500046 kumaresa@gmail.com Abstract The aim of this article is to give a variety of simple and elementary proofs of the fundamental theorem of algebra. proofs of this kind are in [2], pp.464–467 and [7]. A majority of the project has been dedicated to a proof of the theorem, and the remainder is dedicated to discussion about the theorem and it’s origin and relevance to the high school math environment. The Fundamental Theorem of Algebra S. Kumaresan School of Math. Fundamental theorem of algebra. The Fundamental Theorem of Algebra Of the options, the Fundamental Theorem of Algebra was chosen to be investigated. The diﬃculty with it is that it presumes the knowledge of complex analysis, including complex integration and the Cauchy Theorem. Proof of fundamental theorem of algebra. 1 A Fundamental Lemma Throughout our discussion, we let P(z) := P n k=0 a kz You meet this result first in precalculus and Stat. The Fundamental Theorem of Algebra over the Tropical Semiring 4 Acknowledgments 11 References 11 1. 2007-03-01 18:56 FUNDAMENTAL THEOREM OF ALGEBRA James T. Smith San Francisco State University According to Gauss’s fundamental theorem of algebra, every nonconstant polynomial p with complex coefficients has a complex root . The entire book [1] is devoted to diﬀerent proofs of the Fundamental Theorem of Algebra. Of course, this argument is usually circular, because most of the standard proofs of the spectral theorem for matrices requires the fundamental theorem of algebra (either by explicitly citing that theorem, or implicitly, by borrowing one of the proofs given here, e.g. Contents 1. prove the Fundamental Theorem of Algebra over the tropical semiring. Introduction 1 2. Introduction Tropical geometry was founded by Brazilian mathematician and computer sci-entist Imre Simon. The fundamental theorem of algebra states that every nonconstant polynomial with complex coefficients has a complex root.In fact, every known proof of this theorem involves some analysis, since the result depends on certain properties of the complex numbers that are most naturally described in topological terms.. Our proof is based on a similar idea Proof 2) Next we need to ... And now, to prove the Fundamental Theorem of Algebra using Liouville’s Theorem: Theorem: if is a non constant polynomial with coefficients in , then is not a prime polynomial. The main argument in this note is similar to [2]. ity of polynomials and the extreme value theorem for continuous functions. We may, therefore, define the order of the origin O with respect to the function f(z) for any closed curve C as the net number of complete revolutions made by an arrow joining O to a point on the curve Γ traced out by the point representing f(z) as z traces out the curve C . 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