First order sentence true in $\mathbb{Q}$ but not in $\mathbb{R}$.

Question

I have the following assignment question:

Find a first order sentence that is true in $\langle\mathbb{Q},+,\cdot ,0,1\rangle$ but not in $\langle \mathbb{R},+,\cdot,0,1\rangle$.

Most of what I can think of is second order (such as completeness, or cardinality).

I have the following candidate, though:

$$\forall x\exists y\exists z \quad \Bigg(x=\bigg(\sum_{j=0}^y 1\bigg)\cdot i\bigg(\sum_{j=0}^z 1\bigg)\Bigg).$$

Where $i(\alpha)$ denotes the multiplicative inverse of $\alpha.$

I'm just not sure that this idea of summing "$y$ times" or summing "$z$ times" is actually doable in first order logic. I'm also not sure if somehow I need to clarify that $y$ and $z$ need be integers (and if that's doable in FOL).

Any corrections or clarifications to my candidate are welcome, as well as maybe another example I'm just not thinking of.

Answer 1

Hint: The square root of $2$ is irrational.

Answer 2

Hint:

Think about roots, do they always exist in $\mathbb Q$ and do they always exist in $\mathbb R$?

Well, square roots don't have to exist, but what about other orders?