[FOM] Number theory proof mentioned by Frege

[Rob Arthan]

On 16 Apr 2008, at 17:36, 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?
>

Conway and Smith's excellent "On Quaternions and Octonions" gives a 
very accessible account of this and the generalisations when the laws 
are relaxed to drop commutativity and then associativity.

Regards,

Rob.