Can the ordering of the rationals be defined from their field structure alone?

Question

Consider the structure $(\mathbb{Q},+,*,-,0,1)$. Can the canonical ordering of the rational field be defined by a formula from their field structure alone? If yes, can someone give me a formula that defines it?

Answer 1

$$\exists z_1 \exists z_2 \exists z_3 \exists z_4 (y-x=z_1^2+z_2^2+z_3^2+z_4^2)$$