Subject: Re: [HM] ... and the unique factorization theorem
From: Abe Shenitzer (shenitze@pascal.math.yorku.ca)
Date: Sun Feb 13 2000 - 03:15:35 EST
1. The fundamental theorem on finite abelian groups was proved by Ernst
Schering in 1869 and by Leopold Kronecker in 1870 (see, e.g., van der
Waerden's "A history of algebra", pp.149-150). Gauss defined an operation
of composition of classes of forms and showed - in modern terms - that
"the finite set of classes of forms constitutes a (finite) abelian group
under the composition rule" (for a detailed account see Hans Wussing, "The
genesis of the abstract group concept", The MIT Press, 1984, p.55 ff).
Again using modern terms, we can say that Gauss was aware that the
group in question need not be cyclic.
2. The quote referred to by Don is not due to Camille Jordan but to Paul
Gordan.
Abe Shenitzer
------------------------------
On Fri, 11 Feb 2000, Don Cook wrote:
> I was surprised to hear that Gauss is credited with proving three of the
> four fundamental theorems - arithmetic, algebra and Abelian groups (is this
> true?). The Fundamental Theorem of Arithmetic is used before Gauss. Why the
> long wait for a proof? Are there existence theorems proven before Gauss?
>
> Is my suggestion just pop-history? I would like to keep repeating it.
>
> The proof of the Hilbert Basis Theorem is also an existence proof.
> Camille Jordan's remark:
> The proof of the Hilbert Basis Theorem is not mathematics; it is theology.
> Quoted in D MacHale, Comic Sections (Dublin 1993)
>
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