代数基本定理的一个初等代数证明
A Purely Algebraic Proof of the Fundamental Theorem of Algebra
摘要：
代数代数基本定理的证明可以根据所涉及的技术分为三类：依赖于实分析或复分析的证明，代数证明和拓扑证明。代数证明利用奇数次实多项式有实根的事实。然而，这个假设需要解析方法，即实连续函数的中值定理。本文将实多项式中值定理的代数证明思想进一步发展为纯代数证明。在我们的证明中，我们既没有使用连续函数的概念，也没有引用任何实分析和复分析的定理。相反，我们应用现代代数的技术：我们通过超积构造将实数域扩展到超实数的非阿基米德域，并探索有限超实数的子环，无穷小的极大理想和实数之间的关系。
英文摘要：
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 fact that odd-degree real polynomials have real roots. This assumption, however, requires analytic methods, namely, the intermediate value theorem for real continuous functions. In this paper, we develop the idea of algebraic proof further towards a purely algebraic proof of the intermediate value theorem for real polynomials. In our proof, we neither use the notion of continuous function nor refer to any theorem of real and complex analysis. Instead, we apply techniques of modern algebra: we extend the field of real numbers to the non-Archimedean field of hyperreals via an ultraproduct construction and explore some relationships between the subring of limited hyperreals, its maximal ideal of infinitesimals, and real numbers.
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