AI 中文总结
本文运用代数方法，结合伽罗瓦理论与实闭域概念，证明实闭域F对应的F(√-1)代数闭，进而说明ℝ(√-1)=ℂ的代数闭性，为代数基本定理提供代数视角的证明。
AI 中文摘要
本文研究代数基本定理的代数对应物，探讨实闭域与二次型的概念；通过伽罗瓦理论证明，若F为实闭域，则F(√-1)是代数闭的；最后通过证明ℝ的实闭性，说明ℝ(√-1)=ℂ的代数闭性。
英文摘要
In this paper, we investigate the algebraic counterpart of the Fundamental Theorem of Algebra. We explore the concept of real closed fields and quadratic forms. We show, by means of Galois theory, that $F(\sqrt{-1})$ is algebraically closed if $F$ is real-closed. Lastly, we explain the algebraic closure of $\mathbb R(\sqrt{-1})=\mathbb C$ by demonstrating the real-closeness of $\mathbb R$.
Journal refRendiconti del Circolo Matematico di Palermo, Vol 73, 3211 - 3215 (2024)
DOI：10.1007/s12215-024-01093-5
