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even by the standards of modern mathematics? Or are there some mistakes or errors?

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In spite of the fact that his 1849 article is where, for the first time, the Fundamental Theorem of Algebra is stated for polynomials with complex coefficients, the 1849 proof is just his 1799 proof, and therefore it has the same flaw as that proof: Gauss assumes, without proof, that “an algebraic curve can neither suddenly end abruptly anywhere, nor lose itself, so to speak, in a point after an infinity of circuits (as in the case of a logarithmic spiral)” (I am quoting from his dissertation).

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