[FOM] Number theory proof mentioned by Frege

Alasdair Urquhart urquhart at cs.toronto.edu
Thu Apr 17 10:43:03 EDT 2008

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?

More information about the FOM mailing list