What is so special about the real and complex numbers?

Question

When I was studying linear algebra, the first thing we were introduced was the idea of fields. In studying analysis (and when studying inner product spaces etc), we restricted our possible fields to that of reals or complexes. So... here is my question.

1. In defining the inner product space, one restricts the underlying field to either real or complex. It seems to me that in defining the inner product space, it was only necessary that the field be totally ordered. Ah, I forgot, there is this conjugate symmetric property. So, how do we characterize the field so that "conjugacy" makes sense? (involution perhaps?)

2. Okay, how about the normed vector space? I guess, from the name normed vector space, one can define the norm on any totally ordered field? How about the Banach space? By adding the completeness (analytic) condition, can we say that the Banach space can be defined on any totally ordered, complete field?

3. Lastly, as the title suggests, what is so special about the real and complex numbers? Okay, I do know certain properties of these fields, viz. reals are complete, totally ordered, dense, infinite in both directions, etc. But do there exist fields that are NOT isomorphic to these fields (real and complex) such that they behave EXACTLY the same as the real and complex?

Thanks in advance!

Answer 1

In characteristic zero Q is the smallest field (in the sense that this is contained as a subfield in all of them). And when we complete it (analytically) we get the real numbers and when we take the algebraic closure of the latter we get C. In this sense real numbers and complex numbers are the smallest field to do analysis.

Answer 2

A bit of a wider view: if one wants to work over a field of characteristic zero, one is forced to start with the rationals, as already mentioned. To do analysis one needs a metric that is coupled with the field operations. The usual way to achieve this is by taking an absolute value function on the field. For the rationals all absolute value functions are known: they are the ordinary absolute value function and for each prime number the p-adic value function. Now for each of them one can take the completion, thus getting the reals and the p-adic numbers for each p. Next it is desirable to have algebraically closed fields to do function theory for example. In the case of the reals one gets the complex numbers. In the case of the p-adics however the algebraic closure is not complete anymore. Therefore to get a field, in which one can do analysis, one has to take completion again, which then yields a complete algebraically closed field comparable to the complex numbers for each prime p.

After all one can (try to) do analysis either over the reals or over the p-adics, one can (try to) do function theory over the complex numbers or over the fields obtained by the two-step-procedure described above. In particular there are generalizations of for example funtional analysis to the p-adic case. The subject however is quite problematic due to their topological properties, being totally disconnected at the first place to mention here.