相关论文: A short proof of the Fundamental Theorem of Algebr…

We present a simple short proof of the Fundamental Theorem of Algebra, without complex analysis and with a minimal use of topology. It can be taught in a first year calculus class.

Many proofs of the Fundamental Theorem of Algebra, including various proofs based on the theory of analytic functions of a complex variable, are known. To the best of our knowledge, this proof is different from the existing ones.

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…

The Fundamental Theorem of Algebra (FTA) asserts that every complex polynomial has as many complex roots, counted with multiplicities, as its degree. A probabilistic analogue of this theorem for real roots of real polynomials, commonly…

There are several proofs of the Fundamental Theorem of Algebra, mainly using algebra, analysis and topology. In this article, we have shown that the Fundamental Theorem of Algebra can be proved using Nevanlinna's first fundamental theorem…

We give a new proof of the fundamental theorem of algebra. It is entirely elementary, focused on using long division to its fullest extent. Further, the method quickly recovers a more general version of the theorem recently obtained by…

Carl Friedrich Gauss is often given credit for providing the first correct proof of the fundamental theorem of algebra in his 1799 doctoral dissertation. However, Gauss's proof contained a significant gap. In this paper, we give an…

We give a proof for the fundamental theorem of algebra,using the Fredholm index phenomena

The Fundamental Theorem of Algebra (FTA) asserts that every complex polynomial has as many complex roots, counted with multiplicities, as its degree. A probabilistic analogue of this theorem for real roots of real polynomials, sometimes…

Using subresultants, we modify a recent 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…

We present an elementary proof of the fundamental theorem of algebra, following Cauchy's version but avoiding his use of circular functions. It is written in the same spirit as Littlewood's proof of 1941, but reduces it to more elementary…

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 proofs. Algebraic proofs make use of the…

The Initial Algebra Theorem by Trnkov\'a et al.~states, under mild assumptions, that an endofunctor has an initial algebra provided it has a pre-fixed point. The proof crucially depends on transfinitely iterating the functor and in fact…

Two well known facts from elementary number theory are proven by using Bergman spaces.

This paper presents an elementary and direct proof of the Fundamental Theorem of Algebra, via Bolzano-Weierstrass Theorem on Minima, that avoids: every root extraction, angle, non-algebraic functions, differentiation, integration, series…

The main aim of the present paper is to represent an exact and simple proof for FLT by using properties of the algebra identities and linear algebra.

This paper shows an elementary and direct proof of the Fundamental Theorem of Algebra, via Bolzano-Weierstrass Theorem on Minima and the Binomial Formula, that avoids: any root extraction other than the one used to define the modulus…

A mathematics student's first introduction to the fundamental theorem of finite fields (FTFF) often occurs in an advanced abstract algebra course and invokes the power of Galois theory to prove it. Yet the combinatorial and algebraic coding…

This paper presents an alternative proof of the Fundamental Theorem of Algebra that has several distinct advantages. The proof is based on simple ideas involving continuity and differentiation. Visual software demonstrations can be used to…

We show that an elementary proof of Fermat's Last Theorem (FLT) exists. Our paper also extends the scope of FLT from integers to all rational numbers.