lo.logic - A few questions on model theory, especially model theory of rings

"The complex numbers are easy to deal with, whereas for the integers it is much harder..."

What you might have heard about is that the complete first-order theory of the complex numbers with just the ring operations -- which is the same as the theory of algebraically closed fields of characteristic zero -- is decidable (i.e. "computable"), as was first proved by Tarski. So in principle one could write a computer program where you could input any first-order sentence in the language of rings, and in a finite amount of time it would tell you whether or not this sentence is true in the complex numbers.

However, if you look at the ring of integers, no such thing is true; the first-order theory of the integers is undecidable. This is a theorem of Alonzo Church, and is closely related to Goedel's famous incompleteness theorem.

The negative answer to Hilbert's Tenth Problem is a different issue -- this doesn't follow immediately from Church's Theorem, and was proved much later, by Davis, Putnam, Julia Robinson, and Matiyasevich.

I think of these things as more "logical folklore" than model theory, per se -- any reasonable introductory book on mathematical logic (e.g. Enderton's A Mathematical Introduction to Logic) will have a lot to say about them.