# The Fundamental Theorem of Algebra and Linear Algebra

@article{Derksen2003TheFT, title={The Fundamental Theorem of Algebra and Linear Algebra}, author={Harm Derksen}, journal={The American Mathematical Monthly}, year={2003}, volume={110}, pages={620 - 623} }

The first widely accepted proof of the Fundamental Theorem of Algebra was published by Gaus in 1799 in his Ph.D. thesis, although to current standards this proof has gaps. Argand gave a proof (with only small gaps) in 1814 which was based on a flawed proof of d’Alembert of 1746. Many more proofs followed, including three more proofs by Gaus. For a more about the history of the Fundamental Theorem of Algebra, see [5, 6]. Proofs roughly can be divided up in three categories (see [3] for a…

## 23 Citations

A Simple Complex Analysis and an Advanced Calculus Proof of the Fundamental Theorem of Algebra

- MathematicsAm. Math. Mon.
- 2009

Two proofs of the Fundamental Theorem of Algebra which do not seem to have been observed before and which the author thinks are worth recording are presented.

A Purely Algebraic Proof of the Fundamental Theorem of Algebra

- Mathematics
- 2015

Proofs of the fundamental theorem of algebra can be divided up into three groups according to the techniques involved: proofs that rely on real or complex analysis, algebraic proofs, and topological…

A Nonstandard Proof of the Fundamental Theorem of Algebra

- MathematicsAm. Math. Mon.
- 2005

This area of study, model theory, shows that for any consistent theory there are many models, or universes, that can be used as interpretations for its symbols.

Formalized algebraic numbers: construction and first-order theory. (Formalisation des nombres algébriques : construction et théorie du premier ordre)

- Mathematics
- 2012

A formalization of algebraic numbers and their theory is presented and in Coq/SSReflect a framework to work with quotient types is provided and a complete library about ordered and normed number algebraic structures is provided.

Addendum to: "An algebraic proof of the fundamental theorem of algebra"

- Mathematics
- 2018

We prove the fundamental theorem of algebra (FTA on brief) by using linear algebra. The proof which arises from a new equivalent reformulation of FTA also works for any infinite field, having root…

Complex Odd-Dimensional Endomorphism and Topological Degree

- Mathematics
- 2015

A linear mapping upon real n-dimensional space, where the dimension n is odd, has a real eigenvalue-eigenvector pair. The corresponding statement for complex vector spaces holds true for any…

A Machine-Checked Proof of the Odd Order Theorem

- MathematicsITP
- 2013

This paper reports on a six-year collaborative effort that culminated in a complete formalization of a proof of the Feit-Thompson Odd Order Theorem in the Coq proof assistant, using a comprehensive set of reusable libraries of formalized mathematics.

A linear algebra proof of the fundamental theorem of algebra

- Mathematics
- 2010

We present a recent proof due to Harm Derksen, that any linear operator in a complex finite dimensional vector space admits eigenvectors. The argument avoids the use of the fundamental theorem of…

Proving the existence of eigenvalues and eigenvectors by Weierstrass's theorem

- Mathematics
- 2011

The vector v̄ is then an eigenvector with associated eigenvalue λ̄. This theorem is the first step in the study of spectral decomposition and invariant subspaces of T . The traditional proof that I…

True-pairs of Real Linear Operators and Factorization of Real Polynomials

- Mathematics
- 2021

A linear operator on a finite dimensional nonzero real vector space may not have an eigenvalue. However, corresponding to each such operator ) , there exist a pair of real numbers (U, V) and a…

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