An algebraic approach to the Fundamental Theorem of Algebra

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 \(\mathcal {F}(\sqrt{-1})\) $\mathcal{F}(\sqrt{-1})$ is algebraically closed if \(\mathcal {F}\) $\mathcal{F}$ is real-closed. Lastly, we explain the algebraic closure of \(\mathbb {R}(\sqrt{-1})=\mathbb {C}\) $\mathbb{R}(\sqrt{-1})=\mathbb{C}$ by demonstrating the real-closeness of \(\mathbb {R}\) $\mathbb{R}$ .