Introduction and Historical Remarks
The Fundamental Theorem of Algebra states that any complex polynomial must have a complex root. This basic result, whose first accepted proof was given by Gauss, lies really at the intersection of the theory of numbers and the theory of equations, and arises also in many other areas of mathematics. The purpose of these notes is to examine three pairs of proofs of the theorem. The first proof in each pair is fairly straightforward and depends only on what could be considered elementary mathematics. However, each of these first proofs lends itself to generalizations that in turn lead to more general results from which the Fundamental Theorem can be deduced as a direct consequence. These general results constitute the second proof in each pair.
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