Fields and Field Extensions

Abstract

We have now given two proofs of the Fundamental Theorem of Algebra. Both of these involved much more analysis (calculus) than algebra. The first relied on the analytic properties of two-variable real-valued functions from advanced calculus as well as the continuity of real polynomials while the second proof followed from the theory of complex analysis. We now turn to a more algebraic approach to the Fundamental Theorem of Algebra. Eventually we will prove, in the language of this approach, that the complex number field ℂ is an algebraically closed field, a concept equivalent to the fundamental theorem.