csc@watmath.UUCP (Computer Sci Club) (02/21/84)
The question was asked are there N-dimensional numbers in the
same way that the complex numbers are 2-dimensional. The answer
depends on what you mean by "numbers". If you mean fields
then the answer is no.
A field F is a finite extention of the reals R if there
exist a1,a2, ... aN elements of F such that any element f of F
can be written as f= r1*a1 + r2*a2 + ... + rN*aN with
r1, ... rN elements of R. The only finite extention of the reals
is C the complex numbers. Sketch of proof (see Herstein, Topics
in Algebra, chapter 7, (read chapters 1-6 first))
If F is a finite extention of R then any element f of F is
a root of a polynomial in R. (basic result of field theory).
All roots of polynomials in R are complex numbers.
Therefore f is a complex number, therefore F is contained
in C (and can be shown equal to C).
There are field extentions of R (fields which contain a copy
of R), for example the field of rational functions p(x)/q(x)
where p and q are real polynomials and q is non zero. But
these extentions are infinite dimensional.
If we relax our definitions of numbers to include division rings
(like fields but ab=ba is not necessarily true) then we get one
more finite extention of the reals the real quaternions. (They
look like a + b*i + c*j + d*k and are four dimensional over the reals)
(This is the only algebraic division ring extention, I am not sure
if it is the only finite division ring extention)
William Hughes