From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math,sci.logic,sci.physics
Subject: Re: Extending the Fundamental Theorem of Algebra
Date: 26 Mar 1998 18:55:31 GMT

In article <6fdlgm$f56$1@nnrp1.dejanews.com>,  <oholden@idirect.com> wrote:
>The FTA states that any polynomial of degree n, with complex
>coefficients has n complex roots. This theorem is valid only
>within the complex normal field. When we expand the number
>system to e.g. four dimensions then the theorem will be:
>
>Any polynomial of degree n, with hypercomplex (4-D) coefficients
>has  n^2  hypercomplex (4-D) roots.

I don't know what definition of "hypercomplex" you're using, but this
is false in both rings  M_2(R) (the ring of 2x2 real matrices) and
H (the ring of quateriona), which are the only two simple, real,
four-dimensional algebras. The equation X^2+1=0 has infinitely many
roots in each case, which is clear since if  x  is any solution and
y  is any invertible element of this ring, then  y^(-1) x y  is another
solution; they're nearly all distinct, too.

I have no problem with people making conjectures, but rather than
stating what the theorem "will be", it would make a lot more sense to
learn some more about the topic first. The nearest university library
should surely have some textbooks on noncommutative rings and
algebras, which is what you seem to be interested in.

dave

(Associative) Noncommutative Ring Theory:
	index/16-XX.html