Re: [HM] The Fundamental Theorem of Algebra
Robert Mena (rmena@csulb.edu)
4 Nov 98 14:05:16 -0500 (EST)
On the purely algebraic proof of the fundamental theorem of algebra: I
agree that you need a small ingredient as mentioned by Martin Davis in
his comment: a real polynomial of odd degree has to have a real root,
a fact proven by Euler. Gauss' second proof uses only that fact as
proven in van der Waerden's History of Algebra (by the way he only
gives four proofs in there). The second proof of Gauss is based on
ideas of Euler and Laplace (and Lagrange) and uses the symmetric
functions on n-variables (I personally like it very much). The only
nonrigorous point in Gauss's proof is that he needed the existence of
a field where the roots definitely existed and he did not have ideal
theory to provide that existence (so instead he agonizes for four
pages justifying the existence of the roots somewhere).