Gauss's second proof of the fundamental theorem of algebra

Another new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree

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What are these?
[12 Feb 2009]
by Carl Friedrich Gauss (1815); the Latin original appears in Volume 3, pages 33-56, of his collected works.
Any polynomial of even degree m is transformed into one of degree 1/2m(m−1); notice that, although this is typically a larger number, it contains one fewer factor of 2. Each root of the derived polynomial determines a pair of roots of the original one via a quadratic equation. Any odd-degree equation has a real root.
This English translation was made by Paul Taylor in December 1983 and corrected by Bernard Leak. A summary of the proof, together with a note by Martin Hyland on its logical significance, appeared in Eureka 45 (1985). The LATEX version was produced in August 2003. Thanks to Mark Wainwright for finding my notes in an old box of papers in Cambridge and returning them to me.

This is www.PaulTaylor.EU/misc/gauss-web.php and it was derived from non_cs/gauss-web.tex which was last modified on 2 June 2007.