I have some trouble with your article, since some of the symbols you use show up differently in different fonts.
For example, the symbol I think you're using for a right arrow shows up as a little R with a circle around it.
My first comment is on your choice of norm -- the
Field Norm has somewhat nicer properties for this context. It is simply the product of the conjugates of the number:
N(a + bc) = a
2 - b
2 c
2
It's nicer because:
(1) It takes values in your base field.
(2) It's a polynomial in the components of your number.
And it has the same order properties as your norm (N(p) < N(q) iff ||p|| < ||q||), so I don't think it changes anything you've done.
Also, it might be useful to treat Q(c) as a vector space over Q. (Again, borrowing from what can be done in complex analysis, or more importantly, what is done in Lorentzian geometry)
This treatment, I think, would clean up some of your definitions, and generalize them too. (you don't have to worry about things with zero or negative norm)
For example, if we write z = x + y c, then we have the differential of a function f is:
<br />
df = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy<br />
And we have the two standard differentials: dz = dx + c dy and d\bar{z} = dx - c dy.
Since these form a basis, we can actually rewrite a generic differential as:
<br />
df = \frac{\partial f}{\partial z} dz + \frac{\partial f}{\partial \bar{z}} d\bar{z}<br />
And then decree a function to be "differentiable" with respect to z if and only if \frac{\partial f}{\partial \bar{z}} = 0.
Notice that ordinary complex differentiation can be treated in exactly this way.
The nifty thing about this treatment is that it's still a reasonable thing to do in more abstract settings. For example, there is still a reasonable way to define a differential of an algebraic function on, say, a finite field of characteristic 7, and all of the above carries over without any problem.
Another useful thing to look at are rings of formal power series over R. For an arbitrary ring R, one defines R[[x]] to be:
<br />
R[[x]] := \left\{ \sum_{i = 0}^{\infty} a_i x^i | a_i \in R \right\}<br />
whether they converge or not. (Though, in the current context, I suspect there's a reasonable way to define a subring of "convergent" formal power series)
Derivatives on such a thing are defined termwise.
The trick with dealing with these sort of things is to avoid doing anything that asks what the "value" of the sum is.
There is also, of course, the field of formal laurent series over R. I think it's denoted R((x)). There is a slight deficiency, though -- they cannot be "left infinite" sums. Convergence really is needed to deal with doubly infinite sums.
I suspect you might enjoy learning algebraic geometry!
