A Purely Algebraic Proof of the Fundamental Theorem of Algebra

Piotr Błaszczyk

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia (2016)

  • Volume: 8, page 7-23
  • ISSN: 2080-9751

Abstract

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Proofs of the fundamental theorem of algebra can be divided upinto three groups according to the techniques involved: proofs that rely onreal or complex analysis, algebraic proofs, and topological proofs. Algebraicproofs make use of the fact that odd-degree real polynomials have real roots.This assumption, however, requires analytic methods, namely, the intermediatevalue theorem for real continuous functions. In this paper, we developthe idea of algebraic proof further towards a purely algebraic proof of theintermediate value theorem for real polynomials. In our proof, we neither usethe notion of continuous function nor refer to any theorem of real and complexanalysis. Instead, we apply techniques of modern algebra: we extend thefield of real numbers to the non-Archimedean field of hyperreals via an ultraproductconstruction and explore some relationships between the subringof limited hyperreals, its maximal ideal of infinitesimals, and real numbers.

How to cite

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Piotr Błaszczyk. "A Purely Algebraic Proof of the Fundamental Theorem of Algebra." Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia 8 (2016): 7-23. <http://eudml.org/doc/296264>.

@article{PiotrBłaszczyk2016,
abstract = {Proofs of the fundamental theorem of algebra can be divided upinto three groups according to the techniques involved: proofs that rely onreal or complex analysis, algebraic proofs, and topological proofs. Algebraicproofs make use of the fact that odd-degree real polynomials have real roots.This assumption, however, requires analytic methods, namely, the intermediatevalue theorem for real continuous functions. In this paper, we developthe idea of algebraic proof further towards a purely algebraic proof of theintermediate value theorem for real polynomials. In our proof, we neither usethe notion of continuous function nor refer to any theorem of real and complexanalysis. Instead, we apply techniques of modern algebra: we extend thefield of real numbers to the non-Archimedean field of hyperreals via an ultraproductconstruction and explore some relationships between the subringof limited hyperreals, its maximal ideal of infinitesimals, and real numbers.},
author = {Piotr Błaszczyk},
journal = {Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia},
keywords = {fundamental theorem of algebra; continuity; real closed field; intermediate value theorem; ultrapower},
language = {pol},
pages = {7-23},
title = {A Purely Algebraic Proof of the Fundamental Theorem of Algebra},
url = {http://eudml.org/doc/296264},
volume = {8},
year = {2016},
}

TY - JOUR
AU - Piotr Błaszczyk
TI - A Purely Algebraic Proof of the Fundamental Theorem of Algebra
JO - Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
PY - 2016
VL - 8
SP - 7
EP - 23
AB - Proofs of the fundamental theorem of algebra can be divided upinto three groups according to the techniques involved: proofs that rely onreal or complex analysis, algebraic proofs, and topological proofs. Algebraicproofs make use of the fact that odd-degree real polynomials have real roots.This assumption, however, requires analytic methods, namely, the intermediatevalue theorem for real continuous functions. In this paper, we developthe idea of algebraic proof further towards a purely algebraic proof of theintermediate value theorem for real polynomials. In our proof, we neither usethe notion of continuous function nor refer to any theorem of real and complexanalysis. Instead, we apply techniques of modern algebra: we extend thefield of real numbers to the non-Archimedean field of hyperreals via an ultraproductconstruction and explore some relationships between the subringof limited hyperreals, its maximal ideal of infinitesimals, and real numbers.
LA - pol
KW - fundamental theorem of algebra; continuity; real closed field; intermediate value theorem; ultrapower
UR - http://eudml.org/doc/296264
ER -

References

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