Subject: Re: [HM] Fundamental Theorem of Algebra
From: Heinz Lueneburg (luene@mathematik.uni-kl.de)
Date: Tue Apr 11 2000 - 09:01:06 EDT
Peter Flor wrote:
>
> I have preferred this proof (without Conway´s normalizations, as I
> have to admit) to all others I know since I first learned it. It is
> frequently called "Argand´s proof", and I stuck to this usage in some
> lectures. Is this attribution historically correct?
>
I and others call this proof the Cauchy-Argand proof of the FTA. I got from
Doerrie the quotation
Argand, Annales de Gergonne 1815.
I have never seen this paper. Cauchy has published his proof in his
Cours d'analyse. Paris 1821. Oeuvres, Ser. 2, Vol 3. Paris 1897
and in
Exercises de mathe/matiques. Quatrie\me anne/e. Paris 1829
Cauchy writes in his Cours d'analyse that his proof is a variant of the proof
Legendre gives in his The/orie des Nombres, .I.re Partie, Sect. XIV. I looked
it up this afternoon. In this section, Legendre developes reel roots of
polynomials in continued fractions the way Lagrange did. Then he showes how to
improve a given approximation to an imaginary root of a polynomial. The step of
improving the approximation must be the step Cauchy used to get the
contradiction that the minimum |f(k)| of the polynomial f inside a certain
circle around 0 can in fact be made smaller if it is not 0. I have not checked
the details.
Legendre, The/orie des Nombres. Paris, an VI. Pages 161 ff.
Sect. XIV starts on page 133.
Best regards, Heinz Lueneburg
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