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 (-1) F (- 1) is algebraically closed if F F is real-closed. Lastly, we explain the algebraic closure of R (-1) = C R (- 1) = C by demonstrating the real-closeness of R R.