The Fundamental Theorem of Algebra states that every nonconstant polynomial with complex coefficients has at least one complex root. Despite its name, this theorem is usually encountered in analysis rather than algebra. The reason is that, until recently, the techniques used in its proof were found almost exclusively in complex analysis or topology.
The classical formulation of the basic theorem of algebra can be reformulated in several ways, some of which are encountered in the opening chapters. In the central part of the master's thesis, we present a proof that is mostly algebraic, but at the same time does not require anything more than knowledge of linear operators and eigenvectors. This proof is not well known and is of recent date. We also present classical proofs with the help of basic concepts of analysis (compactness, continuity) as well as with more sophisticated tools from complex analysis (Liouville's theorem, Cauchy's integral formula), and one topological proof. We conclude the work with a sketch of C. F. Gauss's first proof and its newer version.
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