Let $K/Q$ be an algebraic extension of the rational numbers. J. Robinson has proved that exists a formula $\psi$ such that when applied to the rational numbers, the seperated set is percisely the ring integers (making the theory of Q undecidable, due to Godel's theorem). However, due to Tarski's results, the theory of the real field is complete and therefore cannot define the integers.

I will ask two related questions:

1. Without any further assumptions on the extension, are the integers necessarily a definable set in $<K,+,\times,0,1>$?
2. What *purely algebraic* properties of the extension might yield a more conclusive answer to the above questions? Specifically: seperability, dimension, normality, finiteness, simplicity (maybe even Galois group structure). Moreover, is there a deeper side to this? Do there exist two extensions $K_{1}/Q,K_{2}/Q$ that all satisfy the exacy same algebraic properties denoted above, yet in one of them $\mathbb Z$ is a definable set, and in the other it is not? 

These are not questions on which I have pondered for long, but I am curious whether exist some interesting examples or logical tools needed in odrder to tackle these questions.