An Algebraic Approach to the Fundamental Theorem of Algebra

Abstract

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 $\mathbb{R}$.