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