J. Robinson has proved that exists a formula $\psi(x)$ in the language of rings which,applied to the rational numbers, defines the the ring integers (making the theory of $\mathbb{Q}$ undecidable, due to Godel's theorem). However, due to Tarski's results, the theory of the real field is complete and therefore the ring of integers $\mathbb{Z}$ is not definable in $\mathbb{R}$.

Let $K/\mathbb{Q}$ be an algebraic extension of the rational numbers. I will ask two related questions:

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

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