From: Harald Hanche-Olsen <hanche@math.ntnu.no>
Newsgroups: sci.math.research
Subject: Re: The Book
Date: 21 Sep 1998 19:39:09 +0200

- Stephen Montgomery-Smith <stephen@showme.missouri.edu>:

| Allen Adler wrote:
| 
| > Here is my proof of the Fundamental Theorem of Algebra (although it
| > has probably occurred to lots of people): let C denote the field
| > of complex numbers and let D be a division algebra of finite
| > dimension n over C. Denote by f the mapping from D-{0} to itself
| > which associates to each nonzero element z its reciprocal 1/z in D.
| > Then f is holomorphic. If n>1 then the isolated singularity 0 of f is
| > removable .........................
| 
| How do you show this last step, that if n>1 then the singularity
| is removable?

This is a standard result in the theory of several complex variables.
In fact, it is a corollary of Hartog's phenomenon:  Let A be an
annulus in the complex plane, and D  the disk which is obtained from A
by filling in the hole.  Let U be an open, connected subset of C^{n-1}
and V a nonempty open subset of U.  If f is holomorphic in the union
of AxU and DxV then f has a holomorphic extension to DxU.

The proof is essentially to write f(w,z)=sum a_n(z)w^n where z in U, w
in A.  For general z you expecte the full Laurent series to be needed,
but when z is in V it follows from the assumptions that the series is
a Taylor series, i.e., a_n(z)=0 when n<0.  By analytic continuation,
a_n is identically zero in U for n<0.

-- 
* Harald Hanche-Olsen     <URL:http://www.math.ntnu.no/~hanche/>
- ��There arises from a bad and unapt formation of words
   a wonderful obstruction to the mind.��  - Francis Bacon