An algebraic proof of the fundamental theorem of algebra

2017 ◽  
Vol 11 ◽  
pp. 343-347
Author(s):  
Ruben Puente
Keyword(s):  
Fundamental Theorem ◽  
Algebraic Proof ◽  
Fundamental Theorem Of Algebra

A NOTE ON THE FUNDAMENTAL THEOREM OF ALGEBRA

2018 ◽  
Vol 97 (3) ◽  
pp. 382-385
Author(s):  
MOHSEN ALIABADI
Keyword(s):  
Real Number ◽  
Fundamental Theorem ◽  
Real Root ◽  
Nonnegative Real Number ◽  
Square Root ◽  
Algebraic Proof ◽  
Real Numbers ◽  
Prime Degree ◽  
Fundamental Theorem Of Algebra ◽  
Odd Degree

The algebraic proof of the fundamental theorem of algebra uses two facts about real numbers. First, every polynomial with odd degree and real coefficients has a real root. Second, every nonnegative real number has a square root. Shipman [‘Improving the fundamental theorem of algebra’, Math. Intelligencer29(4) (2007), 9–14] showed that the assumption about odd degree polynomials is stronger than necessary; any field in which polynomials of prime degree have roots is algebraically closed. In this paper, we give a simpler proof of this result of Shipman.

scholarly journals Quantitative fundamental theorem of algebra

2019 ◽  
Vol 70 (3) ◽  
pp. 1009-1037 ◽  
Author(s):  
Daniel Perrucci ◽  
Marie-Françoise Roy
Keyword(s):  
Fundamental Theorem ◽  
Quantitative Information ◽  
Algebraic Proof ◽  
Intermediate Value Theorem ◽  
Real Polynomials ◽  
Fundamental Theorem Of Algebra ◽  
Quantitative Results ◽  
The Difference ◽  
Intermediate Value ◽  
Classical Proof

Abstract Using subresultants, we modify a real-algebraic proof due to Eisermann of the fundamental theorem of Algebra (FTA) to obtain the following quantitative information: in order to prove the FTA for polynomials of degree d, the intermediate value theorem (IVT) is required to hold only for real polynomials of degree at most d2. We also explain that the classical proof due to Laplace requires IVT for real polynomials of exponential degree. These quantitative results highlight the difference in nature of these two proofs.

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