[FOM] Number theory proof mentioned by Frege

Chris Gray cpgray at library.uwaterloo.ca
Fri Apr 18 09:28:26 EDT 2008


Thanks for your pointer.  It turns out that Remmert, in the chapter on the 
Fundamental Theorem of Algebra, covers the same passage in Hankel's book 
that Frege mentions.


On Thu, 17 Apr 2008, Alasdair Urquhart wrote:

> The book "Numbers" by Ebbinghaus et al. (Springer Graduate
> Texts in Mathematics, Volume 123) has an excellent discussion of
> hypercomplex number systems, written by Koecher and Remmert,
> including historical material.
> In particular, it includes
> a detailed proof of Frobenius's theorem of 1877 that there are
> exactly 3 non-isomorphic real finite-dimensional associative
> division algebras, namely R, C and H (Hamilton's quaternions).
> The result was proved independently by Charles Sanders Peirce
> in 1881.  The result quoted by Frege seems to be a corollary
> to Frobenius's result.
> Alasdair Urquhart
> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
> Chris Gray wrote:
>> In "The Foundations of Arithmetic", Frege mentions a proof from pp.106-7
>> of Hermann Henkel's Theorie der complexen Zahlensysteme.
>> "Hankel proves that any closed field of complex numbers of higher order
>> than the ordinary, if made subject to all the laws of addition and
>> multiplication, contains a contradiction."  Frege p. 106
>> I have no copy of the Henkel book available to me.  Is this a well-known
>> result?  Is this proof discussed or given elsewhere?
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