Proof of the Fundamental Theorem of Algebra

Wolfenstein, Samuel

doi:10.2307/2315822

Cited by 9 publications

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“…It is interesting to note in this connection that there does exist a proof of the FTA which is based on a fixed point theorem (the Lefschetz fixed point theorem; see [11]). In addition, a careful reading of the proof of the FTA given in [21] shows that one of its ingredients is an argument similar to the one appearing in the proof of the Banach fixed point theorem.…”

Section: Introductionmentioning

confidence: 99%

“…For example, R. L. Thompson [16] proved that the FTA is equivalent to the open mapping theorem for polynomials. In addition, the elementary proof of the FTA by S. Wolfenstein [21], and the proof of the generalization of the FTA by M. Reichaw (Reichbach) [13, p. 160] are also related to connectedness and open mappings, but their arguments are different from ours. (They use the fact that f is locally open when f ′ (x) = 0, that f ′ (x) = 0 only on a finite set of points A when f is a polynomial, and that C\A is connected.…”

Section: Introductionmentioning

confidence: 99%

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The open mapping theorem and the fundamental theorem of algebra

Reem

2007

Preprint

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This note is devoted to two classical theorems: the open mapping theorem for analytic functions (OMT) and the fundamental theorem of algebra (FTA). We present a new proof of the first theorem, and then derive the second one by a simple topological argument. The proof is elementary in nature and does not use any kind of integration (neither complex nor real). In addition, it is also independent of the fact that the roots of an analytic function are isolated. The proof is based on either the Banach or Brouwer fixed point theorems. In particular, this shows that one can obtain a proof of the FTA (albeit indirect) which is based on the Brouwer fixed point theorem, an aim which was not reached in the past and later the possibility to achieve it was questioned. We close this note with a simple generalization of the FTA. A short review of certain issues related to the OMT and the FTA is also included.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The open mapping theorem and the fundamental theorem of algebra

Reem

2007

Preprint

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“…An elementary proof of (FTA) based on d'Alembert's idea was given by Fefferman [3] and in shorter form by Aigner, Ziegler [1]. There are many other more or less elementary proofs of (FTA), see [2], [4] and [5]. We improve on the proof in [1] by a much shorter and more transparent argument in the key lemma.…”

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confidence: 98%