A Purely Algebraic Proof of the Fundamental Theorem of Algebra

• Authors: Piotr Błaszczyk

Title variants

PL

Czysto algebraiczny dowód zasadniczego twierdzenia algebry

Abstracts

EN

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.

Keywords

EN

fundamental theorem of algebra; continuity; real closed field; intermediate; value theorem; ultrapower

• Publisher: Uniwersytet Komisji Edukacji Narodowej w Krakowie
• Journal: Annales Universitatis Paedagogicae Cracoviensis. Studia ad Didacticam Mathematicae Pertinentia
• Year: 2016
• Volume: 8
• Pages: 7-23

Dates

published

2016

Contributors

author

Piotr Błaszczyk

• Instytut Matematyki, Uniwersytet Pedagogiczny w Krakowie

Identifiers

Biblioteka Nauki

1791024