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I know that Gauss pointed out flaws in Euler and d'Alembert's FTA proof when he submitted the FTA proof as a doctoral thesis in 1799, but I don't think I've ever seen any mention of Lagrange, so I'm asking. Thanks for any help you can provide!

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Yes, Gauss mentions Lagrange too. He wrote “This great mathematician tried above all to fill in the gaps in Euler's first proof, and indeed, as regards what constitutes the second and fourth objections referred to in §8 above, he has pursued his investigations so profoundly that nothing more remains to be desired. […] On the other hand he has not touched at all the third objection; in fact his whole investigation is based on the assumption that every equation of the $m$th degree actually has $m$ roots.”

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