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 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 .