The Fundamental Theorem of Algebra is True -- Robert Lipshitz
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The Fundamental Theorem of Algebra says that every nonconstant complex polynomial has a root. We will sketch eight proofs of this fact, using: calculus, complex analysis, algebraic topology, combinatorics (with a topological gap), smooth topology, complex analysis (again), Galois theory, and the theory of Lie groups. We will conclude by discussing Bezout's Theorem, a generalization of the FTA.
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The first half should be broadly accessible; the second half will become gradually more arcane.