(in)exactness of complex numbers

[Michael Ackerman]

"David C. Ullrich" wrote:

> Exactly what definition of
> "complex number" do you have in mind here?
> The _standard_ definition _is_ "pair of
> real numbers".)

If you define the complex numbers C as pairs of reals, you have given at
most its structure of real vector space. Then you must define
multiplication and prove that what you have is actually a ring. But if
you define C as the quotient ring R[X]/(X^2+1), then you needn't do
anything more. This definition is often used, e.g. in MacLane &
Birhkoff's "Algebra", the most categorically oriented basic algebra
text.  

-- Michael Ackerman